A new approach to Legendre-truncated-exponential-based Sheffer sequences via Riordan arrays⋆

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A new approach to Legendre-truncated-exponential-based Sheffer sequences via Riordan arrays H.M. Srivastava a,b, Mumtaz Riyasat c, Subuhi Khan c, Serkan Araci d,∗, Mehmet Acikgoz e a

Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China c Department of Mathematics, Aligarh Muslim University, Aligarh 202002, Uttar Pradesh, India d Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, Gaziantep TR-27410, Turkey e Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, Gaziantep TR-27310, Turkey b

a r t i c l e

i n f o

Article history: Received 8 April 2019 Revised 30 June 2019 Accepted 18 August 2019 Available online xxx MSC: Primary 11B83 12E10 Secondary 11C20 Keywords: Sheffer sequences Legendre-truncated-exponential-based Sheffer sequences Riordan arrays Determinant approach Differential equations

a b s t r a c t The significance of multi-variable special polynomials has been identified both in mathematical and applied frameworks. The article aims to focus on a new class of 3-variable Legendre-truncated-exponential-based Sheffer sequences and to investigate their properties by means of Riordan array techniques. The quasi-monomiality of these sequences is studied within the context of Riordan arrays. These sequences are expressed in determinant forms by utilizing the relation between the Sheffer sequences and Riordan arrays. Certain special Sheffer sequences are used to construct the Legendre-truncatedexponential-based Chebyshev polynomials. The same polynomials are used as base to introduce the hybrid classes involving the exponential polynomials and the Euler polynomials of higher order as illustrative examples of the aforementioned general class of polynomials. The shapes are shown and zeros are computed for these sequences by using mathematical software. The main purpose is to demonstrate the advantage of using numerical investigation and computations to discover fascinating pattern of scattering of zeros through graphical representations. © 2019 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries Several applications related to the Sheffer sequences can be find in fields of applied mathematics, theoretical physics, approximation theory et cetera. Sheffer polynomial sequences contain the corresponding associated sequences as well as the Appell sequences as two subclasses. Sheffer sequences are studied systematically by theory of modern umbral calculus (see, for example, [15–17]; see also several related recent works including [11,19,20,22]). We recall some preliminaries from the work by Wang [23]. Let K be a field of characteristic zero and suppose that H is the set of all formal power series in the variable m over  k K such that an element of H is h(m ) = ∞ k=0 ak m , where ak ∈ K for all k ∈ N0 := N ∪ {0} = {0, 1, 2, . . .}. The order o(h(m)) ∗

Corresponding author. E-mail addresses: [email protected] (H.M. Srivastava), [email protected] (S. Araci), [email protected] (M. Acikgoz).

https://doi.org/10.1016/j.amc.2019.124683 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

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of a power series h(m) is the smallest integer k for which the coefficient of mk does not vanish. The series h(m) has a  −1 1 multiplicative inverse, denoted by [h(m )] or by h(m ) , if and only if o h(m ) = 0. In this case, h(m) is called an invertible series. The series h(m) has a compositional inverse, denoted by h(m ) and satisfying the following condition:

h(h(m )) = h(h(m )) = m ⇐⇒ o(h(m )) = 1. The series h(m) with o(h(m )) = 1 is called a delta series. Definition 1. For the invertible series l(m) and delta series h(m) defined as:

h (m ) =

∞ 

hn

n=0

mn n!

( h0 = 0 ; h1 = 0 )

l (m ) =

and

∞  n=0

ln

mn n!

( l0 = 0 ),

(1.1)

the exponential generating function for the sequence sn (u) is given by

1 l (h(m ))

eu(h(m )) =

∞ 

sn ( u )

n=0

mn , n!

(1.2)

for all u ∈ C, where h(m ) is the compositional inverse of h(m). The sequence (sn (u ))N0 in Eq. (1.2) is the Sheffer sequence for the pair (l(m), h(m)). The Sheffer sequences for the pair (1, h(m)) reduce to the associated Sheffer sequence and for the pair (l(m), m) reduce to the Appell sequence (see, for details, [16]; see also [21]). The applications of Appell polynomials lie in probability theory and statistics. The generalized Appell polynomials are considered as important tool for approximating 3D-mappings in combination with Clifford analysis methods. These polynomials also occur in representation theory in the field of quantum physics. The concept of the Riordan arrays was introduced by Shapiro et al. [18]. For this, we have Definition 2. A generalized Riordan array with respect to the sequence (cn )n∈N0 is a pair (l(m), h(m)), which is an infinite lower triangular array (an,k )0kn < ∞ satisfying the rule given below:



an,k =



mn [ h ( m )] k l (m ) , cn ck

(1.3)

k where quotients such as l (m )[ch(m )] are called the column generating functions of the Riordan array. When cn = 1, the clask

sical Riordan arrays are obtained and the exponential Riordan arrays correspond to the case when cn = n!. For any fixed sequence (cn )n∈N0 , the set of all Riordan arrays (l(m), h(m)) is a group under matrix multiplication and is called a Riordan group with respect to the sequence (cn )n∈N0 . The identity of this group is (1, m) and the inverse of the array



(l(m), h(m)) is

1 , h (m ) l (h(m ))



, where h¯ (m ) is the compositional inverse of h(m). The Riordan matrices naturally appear in a

formulation of the umbral calculus. The Riordan group also appears in the new domain of combinatorial quantum physics, namely in the problem of normal ordering of boson strings. The Sheffer sequences can also be represented via algebraic (determinant) form [23]. The determinant forms are helpful for several numerical and computation purposes. Let (sn (u ))n∈N0 be Sheffer sequence for the pair (l(m), h(m)) satisfying the following condition:

un =

n 

an,k sk (u ).

(1.4)

k=0

Then an,k is the (n, k) entry of the Riordan array (l(m), h(m)) (see [24]). We now recall the following determinant definition of Sheffer sequence (see [23]). Let (sn (u ))n∈N0 be a Sheffer sequence for the pair (l(m), h(m)). Then we have

s0 ( u ) =

sn ( u ) =

1 a0,0

a0,0



1



a 0 , 0



0

n

(−1 )

a1,1 · · · an,n

0

·

·



0

u

u2

···

un−1

a1,0

a2,0

···

an−1,0

a1,1

a2,1

···

an−1,1

0

a2,2

···

an−1,2

· ·

· ·

··· ···

· ·

0

0

···

an−1,n−1





an,0



an,1



an,2



·

·



an,n−1 un

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(−1 )n Un+1 det , Sn×(n+1) a0,0 a1,1 · · · an,n

= where

3



Un+1 = (1, u, u2 , · · · , un−1 , un )

(1.5)

Sn×(n+1) = (a j−1,i−1 )1in; 1 jn+1

and

and an,k is the (n, k) entry of the Riordan array (l(m), h(m)). The Sheffer sequence (sn (u ))n∈N0 defined by the equation (1.5) also satisfy the following condition:

sn ( u ) =

 where bn,k is the (n, k) entry of the Riordan array

n 

bn,k uk ,

k=0

1 , h (m ) l (h(m ))



(1.6)

and (sn (u ))n∈N0 is a Sheffer sequence for the pair (l(m),

h(m)) (see [24]). In recent years, a remarkable progress has been observed in the multi-variable forms of special functions of mathematical physics. To solve the problems that arise in various branches of mathematics, going from the theory of partial differential equations to the abstract group theory, necessities of multi-index and multi-variable special functions are realized. We recall the following two-variable forms of special polynomials: The Legendre polynomials are closely associated with physical phenomena for which spherical geometry is important. In particular, these polynomials arise in the problem of expressing the Newtonian potential of a conservative force field in an infinite series involving the distance variables of two points and their included central angle. The 2-variable Legendre polynomials (2VLeP) Sn (u, v ) [9] are defined by the following generating function: ∞  √ mn evm J0 (2m −u ) = Sn (u, v ) , n!

(1.7)

n=0

where Jν (um) is the Bessel function of first kind of order ν defined by (see [2,25])

Jν (w ) =

∞  k=0

 w ν +2k (−1 )k k! (ν + k + 1 ) 2

(w ∈ C \ (−∞, 0]; ν ∈ C ).

(1.8)

The Bessel functions arise in the study of free vibrations of a circular membrane and in finding the temperature distribution in a circular cylinder. They also occur in electromagnetic theory and numerous other areas of physics and engineering. The series expansion for the 2VLeP Sn (u, v ) is given by (see [9]): n

Sn (u, v ) = n!

[2] 

vn−2k . ( k ! ) ( n − 2 k )! uk

k=0

We also note that



exp −α

D−1 u



= J0 (2

√



αu)

D−n u



un {1} := is the inverse derivative operator . n!

In view of the Eq. (1.7) and (1.10), it follows that





(1.9)

2

Sn (u, v ) = Hn

(1.10)



 , v, D−1 u

where Hn v, D−1 denotes the 2-variable Hermite-Kampé de Fériet polynomials (see [3]). These polynomials play an imporu tant role in problems involving Laplace equation in parabolic coordinates and are shown to be solutions of classical and generalized heat equations. The generating function for the 2-variable truncated-exponential polynomials (2VTEP) en(r ) (v, w ) of order r is given by (see [7, p. 174, Eq. (30)]): ∞

 (r ) evm mn = en ( v, w ) , r 1 − wm n!

(1.11)

n=0

where

( 1 − w ) −λ =

∞  k=0

( λ )k

wk k!

(1.12)

in terms of the Pochhammer symbol or the shifted factorial (λ )k (k ∈ N0 ), which are involved in various combinatorial problems. The series expansion for the 2VTEP en(r ) (v, w ) of order r is given by

en(r ) (v, w ) = n!

[ nr ]  wk vn−rk . (n − rk )!

(1.13)

k=0

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In this article, a general class of 3-variable Legendre-truncated-exponential-based Sheffer sequences is studied using Riordan array techniques. These sequences are studied within the framework of the monomiality principle and are expressed by means of determinants. In order to show some applications of main results several illustrative examples are also constructed. Certain graphical representations and numerical computations are also presented. 2. Legendre-truncated-exponential-based Sheffer sequences Here, in this section, we first introduce the 3-variable Legendre-truncated-exponential polynomials (3VLeTEP) of order r via operational view point. Operational techniques are applicable to solve problems both in classical and quantum mechanics. Finding the differential equations being the primary motivation for the introduction of these techniques. A systematic and analytic approach to study special functions of one or more variables involves the use of operational techniques including the operators. The multiplicative and derivative operators associated with the polynomials Sn (u, v ) and en(r ) (v, w ) are as follows:

+S = v + 2D−1 u Dv , and

−S = Dv

+e = v + rwDw wDr−1 v ,

(2.1)

−e = Dv .

(2.2)

Further, to introduce new families of special polynomials and to deal with the theoretical foundations of special polynomials, the operational techniques with monomiality principle [5] play a major role. Indeed, upon replacing v by the multiplicative operator + given by the equation (2.1) in both sides of the equation S (1.11), we have



1 1 − wmr







exp (v + 2D−1 u Dv ) m =

∞  mn en(r ) (v + 2D−1 , u Dv , w ) n!

(2.3)

n=0

which by use of the following Crofton-type identity (see [8]):



dm−1 f v + mλ m−1 dv for (m = 2 ) in the l.h.s. becomes



1 1 − wmr



 m d {1} = exp λ m { f (v )} dv

2 exp(D−1 u Dv ){exp (vm )} =

(2.4)

∞  mn en(r ) (v + 2D−1 . u Dv , w ) n!

(2.5)

n=0

Now, on expanding the first exponential with the use of formula (1.12) in the l.h.s. and denoting the resulting 3VLeTEP in the right-hand side by (r ) S en

(u, v, w ) = en(r ) (v + 2D−1 u Dv , w ) ,

(2.6) (r )

we find the following generating equation for the 3-variable Legendre-truncated-exponential polynomials S en (u, v, w ) of order r:



1 1 − wmr



∞  √ mn (r ) evm J0 (2m −u ) = . S en (u, v, w ) n!

(2.7)

n=0

Using Eqs. (1.7) and (1.12) as well as (1.11) and (1.8), respectively, in Eq. (2.7), the following explicit series expansions for 3VLeTEP S en(r ) (u, v, w ) of order r are obtained: n

(r )

S en

[r]  wk Sn−rk (u, v ) (u, v, w ) = n! , (n − rk )!

(2.8)

k=0

n

(r ) S en

(u, v, w ) = n!

[2]  k=0

) uk en(r−2 ( v, w ) k

k! (k + 1 ) (n − 2k )!

,

(2.9)

respectively. Further, in view of expansions (1.9) and (1.13), the above expansions can be expressed as: (r )

S en

(u, v, w ) = n!

n

n−2k

k=0

l=0

[2] [ r ]  

n

n−rk

k=0

l=0

[r] [ 2 ]   wl vn−2k−rl uk ul vn−rk−2l wk = n! . k! (k + 1 ) (n − 2k − rl )! l! (l + 1 ) (n − rk − 2l )!

(2.10)

+ Replacing v in expression (2.2) of multiplicative operator + e by expression (2.1) of multiplicative operator S , we get

the following multiplicative and derivative operators for the 3VLeTEP S en(r ) (u, v, w ): r−1 +Se = v + 2D−1 u Dv + rwDw wDv ,

−Se = Dv ,

(2.11)

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so that we have

r−1 v + 2D−1 u Dv + rwDw wDv



(r ) S en

5

 ) (u, v, w ) = S en(r+1 (u, v, w ),

(2.12)

) Dy {S en(r ) (u, v, w )} = n S en(r−1 (u, v, w ).

(2.13)

(r )

The differential equation satisfied by the 3VLeTEP S en (u, v, w ) is given by



 (r ) 2 r vDv + 2D−1 u Dv + rwDw wDv − n S en (u, v, w ) = 0.

(2.14)

(r )

The 3VLeTEP S en (u, v, w ) are given by the following explicit representation: (r ) S en

    r−1 n (u, v, w ) = (+Se )n {1} = v + 2D−1 {1} S e0(r ) (u, v, w ) := 1 . u Dv + rwDw wDv

(2.15)

Next, we derive the generating equation for the 3-variable Legendre-truncated-exponential-based Sheffer sequences 3VLeTEbSS. Theorem 1. For the pair (l(m), h(m)) and for all u, v, w ∈ C, the generating equation for the 3-variable Legendre-truncatedexponential-based Sheffer sequences of order r is given by



1

l h (m )

 

√

1 1 − w(h(m ))r

 ev(h(m)) J0 (2h(m ) −u ) =

∞  Se

(r )

sn (u, v, w )

n=0

mn . n!

(2.16)

Proof. Consider Eq. (1.2) with u replaced by the multiplicative operator + so that we have Se



1

 e(Se h(m)) = +

l h (m )

∞ 

sn (+ Se )

n=0

mn . n!

(2.17)

Now, using expression of + from Eq. (2.11) and then applying identity (2.4) twice to decouple the exponential operators Se in the l.h.s. of the resulting equation, it follows that



1

l h (m )

∞ n    2 r−1 m sn (v + 2D−1 ,  exp(wDw wDrv ) exp(D−1 u Dv ) exp vh (m ) = u Dv + rwDw wDv ) n=0

n!

(2.18)

which on expanding the second and the first exponential using formulas (1.10) and (1.12) in the l.h.s., and denoting the resulting 3-variable Legendre-truncated-exponential-based Sheffer sequences (3VLeTEbSS) of order r in the r.h.s. by Se

(r )

sn (u, v, w ) = sn (+ Se ) = sn

we are led to the assertion (2.16) of Theorem 1.



 r−1 , v + 2D−1 u Dv + rwDw wDv

(2.19)



Theorem 2. For the pair (l(m), h(m)), the explicit series representation for the 3-variable Legendre-truncated-exponential-based Sheffer sequences of order r is given by Se

(r )

sn (u, v, w ) =

 where bn,k is the (n, k) entry of the Riordan array

n 

bn,k S en(r ) (u, v, w ),

(2.20)

k=0



1

l h (m )

 , h (m ) .

Proof. Replacing u by the multiplicative operator + se of the polynomials 3VLeTEP in both sides of the Eq. (1.6), we find that

sn (+ Se ) =

n 

k bn,k (+ Se ) {1},

(2.21)

k=0

which, on using Eqs. (2.19) and (2.15) in the l.h.s. and the r.h.s., respectively, yields explicit series representation (2.20) asserted by Theorem 2.  2.1. Determinant approach To express the generalized polynomials 3VLeTEbSS

Se

( r ) sn

(u, v, w ) in a determinant form, the following theorem is proved:

Theorem 3. The 3-variable Legendre-truncated-exponential-based Sheffer sequences of degree n are given by Se

(r )

s0 (u, v, w ) =

Se

(r )

1 , a0,0

(2.22)

sn (u, v, w )

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=

=

a0,0

(r ) (r )

1 S e1 (u, v, w ) S e2 (u, v, w )



a a1,0 a2,0

0,0



0 a1,1 a2,1

n

(−1 )

0 0 a2,2 a1,1 . . . an,n



· ·

·

· · ·

· · ·



0 0 0  n (−1 ) SEn+1 (u, v, w )

a0,0 a1,1 . . . an,n

det

(r ) S en−1

··· ···

an−1,0

···

an−1,1

···

an−1,2

··· ··· ···

· · ·

···

an−1,n−1

(r ) S en

(u, v, w )





an,0





an,1



an,2



·



·



·





an,n−1

,

Mn×(n+1)

(2.23)



where

(u, v, w )



SEn+1 (u, v, w ) = 1, S e1(r ) (u, v, w ), . . . , S en(r ) (u, v, w ) , Mn×(n+1) = (a j−1,i−1 )1in,

1 jn+1 ,

an,k being the (n, k) entry of the Riordan array (l(m), h(m)). Proof. Upon replacing u by + in Eq. (1.4) and then appropriately using Eqs. (2.15) and (2.19) in the resulting equation, it Se follows that (r ) S en

(u, v, w ) =

n 

an,k

Se

(r )

sn (u, v, w ),

(2.24)

k=0

where an,k is the (n, k) entry of the Riordan array (l(m), h(m)). In view of identity (2.24), we obtain the following system of infinite equations in the unknowns

⎧ a0,0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a1,0 ⎪ ⎪ ⎪ ⎪ ⎨ a2,0 ⎪ .. ⎪ ⎪ ⎪ ⎪. ⎪ ⎪ ⎪ ⎪an,0 ⎪ ⎪ ⎩.. .

Se

(r )

s0 (u, v, w ) = 1,

Se

(r )

s0 (u, v, w ) + a1,1

Se

(r )

s1 (u, v, w ) = S e1(r ) (u, v, w ),

Se

(r )

s0 (u, v, w ) + a2,1

Se

(r )

s1 (u, v, w ) + a2,2

Se

(r )

s2 (u, v, w ) = S e2(r ) (u, v, w ),

Se

(r )

s0 (u, v, w ) + an,1

Se

(r )

s1 (u, v, w ) + an,2

Se

(r )

s2 (u, v, w ) + . . . + an,n

Se

(r )

Se

( r ) sn

(u, v, w ) (n ∈ N0 ):

(2.25)

sn (u, v, w ) = S en(r ) (u, v, w ).

From the first equation of the system (2.25), the first part of assertion (2.23) is proved. Also, the special form of the system (2.25) (lower triangular) allows us to find the unknowns e(r ) sn (u, v, w ). Operating upon the first n + 1 equations S simply by applying the Cramer’s rule, we are led to

Se

(r )

sn (u, v, w ) =

a0,0



a0,0



a

1,0



a2,0



1

·

a1,1 . . . an,n

·

·





an−1,0



a n,0

0

0

···

0

a1,1

0

···

0

a2,1

a2,2

···

0

· · ·

· · ·

··· ··· ···

· · ·

an−1,1

an−1,2

···

an−1,n−1

an,1

an,2

···

an,n−1







(r ) S e1 (u, v, w )



(r )

e ( u, v , w ) S 2



·

,

·



·



(r ) S en−1 (u, v, w )



(r ) e (u, v, w )

1

(2.26)

S n

which, on bringing the (n + 1 )th column to the first place by means of n transpositions of the adjacent columns and in view of fact that the determinant of a square matrix is same as that of its transpose, completes the proof of second part of assertion (2.23) of Theorem 3.  Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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7

2.2. Quasi-Monomial properties It has been shown that the generalized polynomials 3VLeTEP S en(r ) (u, v, w ) of order r satisfy quasi-monomial properties. With a view to show that the sequences e(r ) sn (u, v, w ) are quasi-monomial, we prove the following result. S

Theorem 4. The multiplicative and derivative operators for the 3-variable Legendre-truncated-exponential-based Sheffer sequences e(r ) sn (u, v, w ) of order r are given by: S



+Ses =

r−1 v + 2D−1 − u Dv + rwDw wDv

and

l ( Dv ) l ( Dv )



1 h ( Dv )



(2.27)

−Ses = h(Dv ).

(2.28)

Proof. In view of the fact that h(m ) is the compositional inverse of the function h(m), the following identity is easily established:

h ( Dv )

⎧ ⎨



v h (m )

⎫ ⎬



√

⎛



v h (m )



⎞



√



1 e 1 e     J0 (2h(m ) −u ) = m ⎝     J0 2h(m ) −u ⎠. ⎩ l h (m ) 1 − w h (m ) r ⎭ l h ( m ) 1 − w h (m ) r

(2.29)

Now, by differentiating Eq. (2.17) partially with respect to m and then using relation (2.19) in the r.h.s. of resulting equation, we find





l h (m )

 −  + Se

l h (m )









1

h h ( m )





1

l h (m )

∞    mn ,  exp +Se h(m ) = (r ) sn+1 (u, v, w ) Se

which, on using the monomiality principle equation exp(m+ p ){1} = nomials S en(r ) (u, v, w ) in the l.h.s. gives





l h (m )

+Se − 

l h (m )

=

∞  Se

(r )



∞

n=0

pn ( u )





⎛

 

1

h h ( m )

sn+1 (u, v, w )

n=0

 ⎝

n!

n=0

1

l h (m )

e



v h (m )

mn n!

(2.30)

and generating Eq. (2.7) for the poly-

⎞



1 − w h (m )



√ J0 (2h(m ) −u )⎠

r

mn . n!

(2.31)

By applying identity (2.29) and generating Eq. (2.16) in the above equation it takes the form:



l ( Dv )  − l ( Dv ) + Se



1 h ( Dv )

 ∞

 n=0

mn (r ) sn (u, v, w ) Se n!



=

∞  Se

(r )

sn+1 (u, v, w )

n=0

mn . n!

(2.32)

Next, on substituting expression for + from Eq. (2.11) in the above equation and then equating the coefficients of same Se powers of m the resulting equation yields assertion (2.27) in view of monomiality equation + { pn (u )} = pn+1 (u ). Next, by using Eq. (2.16) in identity (2.29), we find



h ( Dv )

∞  n=0

mn (r ) sn (u, v, w ) e S n!



=

∞ 

(r ) sn−1 Se

(u, v, w )

n=1

mn . ( n − 1 )!

(2.33)

Finally, by equating the coefficients of same powers of m together with use of monomiality equation − { pn (u )} = n pn−1 (u ), we get assertion (2.28), which completes proof of Theorem 4.  The following consequences of Theorem 4 can be deduced fairly easily. Corollary 1. The 3-variable Legendre-truncated-exponential-based Sheffer sequences of order r satisfy the following differential recurrence relations:

 1 l ( Dv ) r−1 D + rwD wD − v + 2D−1 (r ) sn (u, v, w ) = e(r ) sn+1 (u, v, w ) v w u v S l ( Dv ) h ( Dv ) S e

and

h ( Dv )

Se

(r )

sn (u, v, w ) = n

Se

(r )

sn−1 (u, v, w ).

(2.34)

(2.35)

Corollary 2. The 3-variable Legendre-truncated-exponential-based Sheffer sequences of order r satisfy the following differential equation:  r−1 − v + 2D−1 u Dv + rwDw wDv

l ( Dv ) l ( Dv )

h ( Dv ) −n h ( Dv )

Se

(r )

sn (u, v, w ) = 0.

(2.36)

Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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H.M. Srivastava, M. Riyasat and S. Khan et al. / Applied Mathematics and Computation xxx (xxxx) xxx Table 1 Results for

He

(r )

sn (v, D−1 u , w ).

S. No.

Results

Expressions

I.

Generating equation



1

l h (m )

II.

Series representation

III.

Multiplicative and derivative operators

He

(r )



1

1−w h(m )

v  e

+Hes =









h(m ) +D−1 h (m ) u



)

2

=

r

sn (v, D−1 u , w) =

−Hes  IV.

 

n  k=0

dn,k

(r ) H en

∞  n=0

(r )

sn (v, D−1 u , w)

mn n!



(v, D−1 x , w ); dn,k is the (n, k ) entry of l (D −1 )

−1 + rwDw wDr−1 − l (D u ) D−1 u + 2Dv DD−1 u D−1 u D−1 u ) = h (DD−1 u

1

D

l (D

h(D −1 ) −1 )

−1 − l (D u ) + rwDw wDr−1 D−1 u + 2Dv DD−1 u D−1 u D−1

Differential equation

He

D

Du

h (DD−1 )

u

h ( D

D−1 u

−n



1

l h (m )

 , h (m )



)

Se

(r )

sn (v, D−1 u , w) = 0

u

Corollary 3. The 3-variable Legendre-truncated-exponential-based Sheffer sequences of order r satisfy the following explicit representation:  Se

(r )

sn (u, v, w ) =

r−1 − v + 2D−1 u Dv + rwDw wDv

l ( Dv ) l ( Dv )

1 h ( Dv )

n

{1}.

(2.37)

Remark 1. The 3VLeTEbSS Sn (u, v ) = Hn (v, D−1 u ) of order r becomes the 3-variable Hermite-truncated-exponential-based Sheffer sequences (3VHTEbSS) of order r, which may be denoted by e(r ) sn (v, D−1 u , w ). The corresponding results are preH sented in Table 1. The 3-variable Hermite-truncated-exponential-based Sheffer sequences of order r are defined by He

(r )

He

s0 (v, D−1 u , w) =

(r )

1 a0,0

sn (v, D−1 u , w) =

= where and

a0,0

(2.38)

(r ) −1

1 ··· H e 1 ( v, Du , w )



a a1,0 ···

0,0



0 a1,1 ···

n

(−1 )

0 0 ··· a1,1 . . . an,n



· ···

·

· · ···

· · ···



0 0 ···   −1 HEn+1 (v, Du , w ) (−1 )n

a0,0 a1,1 . . . an,n

Mn×(n+1)

det

(r ) H en−1

(v, D−1 u , w)

an−1,0 an−1,1 an−1,2 · · · an−1,n−1



Mn×(n+1) = (a j−1,i−1 )1in,



(v, D−1 u , w)



an,0





an,1



an,2



·



·



·



an,n−1

,

(r ) (r ) −1 −1 HEn+1 (v, D−1 u , w ) = 1 , H e 1 ( v, Du , w ), . . . , H e n ( v, Du , w )

(r ) H en

(2.39)



1 jn+1 ,

an,k being the (n, k) entry of the Riordan array (l(m), h(m)). To consider the convolution of two or more polynomials in order to introduce new multi-variable generalized polynomials is recent topic of research and is useful from the point of view of application. These polynomials are important as they possess significant properties including the recurrence and explicit relations; functional and differential equations, summation formulae, symmetric and convolution identities, determinant forms et cetera. The usefulness and potential for applications of various properties of multi-variable hybrid special polynomials in certain problems of number theory, combinatorics, classical and numerical analysis, theoretical physics, approximation theory and other fields of pure and applied mathematics is a key factor of motivation for introducing the class of Legendre-truncated-exponential based Sheffer sequences. The properties and applications of these polynomials lie within the parent polynomials. As far as the applications of the hybrid class of Legendre-truncated-exponential based Sheffer sequences concern, we have 1. The hybrid polynomials comprising Legendre polynomials occur in problems dealing with either gravitational potentials or electrostatic potentials. Certain steady-state heat-conduction problems in spherical-shaped solids can also be solved by these hybrid Legendre polynomials. Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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H.M. Srivastava, M. Riyasat and S. Khan et al. / Applied Mathematics and Computation xxx (xxxx) xxx Table 2 Results for

Se

(r )

9

pn (u, v, w ).

S. No.

Results

Expressions

I.

Series representation

II.

Multiplicative and derivative operators

III.

Differential equation

Se

(r )

pn (u, v, w ) =



n 

cn,k S en(r ) (u, v, w ); cn,k is the (n, k ) entry of

 r−1

k=0 2D−1 u Dv





1, h ( m )



1 +Sep = v + + rwDw wDv h ( Dv ) +Sep = h(Dv )    r−1 h (Dv ) v + 2D−1 − n S e(r ) pn (u, v, w ) = 0 u Dv + rwDw wDv h ( Dv )

2. The hybrid polynomials involving Hermite polynomials occur in quantum mechanical and optical beam transport problems and in probability theory. 3. The reason of the interest for the hybrid polynomials related with truncated-exponential polynomials stems from the fact that they appear in the theory of flattened beams, which plays a role of paramount importance in optics and particularly in super-Gaussian optical resonators. 3. Special members In this section, we discuss some special members of the 3VLeTEbSS and 3VHTEbSS, each of order r. The two particular subclasses of the Sheffer sequences are the sequences of the Appell polynomials and the associated Sheffer polynomials, which are discussed in Section 1. To study the subclasses related to the 3VLeTEbSS e(r ) sn (u, v, w ), we consider the following S cases: I. We know that the Sheffer sequence for the pair (1, h(m)) becomes the associated Sheffer sequence pn (u). Therefore, by taking l (m ) = 1, so that

1



l h (m )

 =1

and

l (m ) = 0

in the 3VLeTEbSS e(r ) sn (u, v, w ), we find that the 3-variable Legendre-truncated-exponential-based associated Sheffer seS quence (3VLeTEbASS) e(r ) pn (u, v, w ) related with the pair (1, h(m)) is defined by the following generating function: S





1



1 − w h (m ) The other results for the 3VLeTEbASS Upon setting



Se

r



e



∞  √ mn J0 (2h(m ) −u ) = . (r ) pn (u, v, w ) Se n!

(3.1)

n=0

( r ) pn

(u, v, w ) are given in Table 2.

 an,0 =



v h (m )

mn cn



h (m )

0

c0

=



cn n [m ]1 = δn,0 . c0





in which case the determinant form of sn (u ) n∈N0 reduces to that of pn (u ) n∈N0 ). Consequently by making the same substitution in the right-hand side of determinant form (2.23), we find that the 3VLeTEbASS e(r ) pn (u, v, w ) are defined by S means of the following determinant: Se

Se

(r )

(r )

p0 (u, v, w ) = 1

pn (u, v, w ) =

a1,1

(r )

S e1 (u, v, w )



a1,1





0 (−1 )n



a2,2 . . . an,n

·



·



·



0

(3.2) (r ) S e2

(r ) S en−1

(u, v, w )

···

(u, v, w )

a2,1

···

an−1,1

a2,2

···

an−1,2

· · ·

··· ··· ···

· · ·

0

···

an−1,n−1

(u, v, w )



an,1





an,2

,

·



·



·



a

(r ) S en

(3.3)

n,n−1

where an,k is the (n, k) entry associated with (1, h(m)). Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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H.M. Srivastava, M. Riyasat and S. Khan et al. / Applied Mathematics and Computation xxx (xxxx) xxx Table 3 Results for

Se

(r )

An (u, v, w ).

S. No.

Results

I.

Series representation

II.

Expressions Se

n  k=0

dn,k S en(r ) (u, v, w ); dn,k is the (n, k ) entry of



1 ,m l (m )





r−1 − ll ((DDvv)) +SeA = v + 2D−1 u Dv + rwDw wDv − = D  v  SeA  l ( Dv ) 2 r vDv + 2D−1 u Dv + rwDw wDv − l (Dv ) Dv − n S e(r ) An (u, v, w ) = 0

Multiplicative and derivative operators Differential equation

III.

An (u, v, w ) =

(r )



II. We know that the Sheffer sequence for the pair (l(m), m) becomes the Appell sequence An (u ) taking h(m ) = m, so that

h ( m ) = 1



and

1

l h (m )

=



n∈N0 .

Therefore, by

1 l (m )

in the 3VLeTEbSS e(r ) sn (u, v, w ), we find that the 3-variable Legendre-truncated-exponential-based Appell sequence S (3VLeTEbAS) e(r ) An (u, v, w ) related with the pair (l(m), m) is defined by the following generating function: S



1 1 l (m ) 1 − wmr



∞  √ mn evm J0 (2m −u ) = . (r ) An (u, v, w ) Se n!

The other results for the 3VLeTEbAS e(r ) An (u, v, w ) are given in Table 3. S In the right-hand side of determinant form (2.23), if we set



an,k =



(3.4)

n=0



mn mk n! n−k l (m ) = [m ]l ( m ) = n! k! k!







in which case the determinant form of sn (u ) n∈N0 reduces to that of An (u ) e(r ) An (u, v, w ) are defined by the following determinant definition:

n k



ln−k , (see [4]), we find that the 3VLeTEbAS

n∈N0

S

Se

(r )

A0 (u, v, w ) =

1 l0

(3.5)



1



l

0



0



(−1 )n

(r ) An (u, v, w ) = Se n+1 0

l0

·

·

·



0

(r ) S e1

(u, v, w )

(r ) S e2

l1

(u, v, w )

···

l2

2

l0

1

(r ) S en−1

···

l1

(u, v, w )

ln−1

n−1

···

1

n−1

0

l0

···

· · ·

· · ·

··· ··· ···

· · ·

0

0

···

l0

2

ln−2 ln−3

(u, v, w )





ln



 n

ln−1 1



 n

, ln−2 2



·



·



·



n

(r ) S en

n−1

(3.6)

l1

where an,k is the (n, k) entry associated with (l(m), m). Remark 2. In view of Remark 1, the results for the 3-variable Hermite-truncated-exponential-based associated Sheffer sequence (3VHTEbASS) e(r ) pn (v, D−1 u , w ) and the 3-variable Hermite-truncated-exponential-based Appell sequence (3VHTEH

bAS)

He

( r ) An

(v, D−1 u , w ) can be obtained easily.

4. Illustrative examples In this section, we consider illustrative examples of certain members belonging to the class of 3VLeTEbSS in order to give applications of the results derived above.

Se

( r ) sn

(u, v, w )

Example 1. In the left-hand side of the generating function (2.16), if we take



l (m ) =



2 , √ 1 + 1 − m2

h (m ) = −

m √ 1 + 1 − m2

and

2m h¯ (m ) = − , 1 + m2

Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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H.M. Srivastava, M. Riyasat and S. Khan et al. / Applied Mathematics and Computation xxx (xxxx) xxx Table 4 Results for

Se

(r )

Un (u, v, w ).

S.No.

Results

I.

Series representation

II.

III.

11

Expressions Se

(r )

[n/ 2]

n−l

) 2n−2l S en(r−2 (u, v, w ) l l  r−1 v + 2D−1 − √ 2 1√ = u Dv + rwDw wDv

Un (u, v, w ) =

(−1 )l

l=0

Multiplicative and derivative operators

+SeU

Differential equation

−Dv −SeU = √  1+ 1−D2v r−1 v + 2D−1 − √ u Dv + rwDw wDv

1−Dv (1+

1√ 1−D2v (1+ 1−D2v )



1−D2v )



−

(1+

√

1−D2v )2 (1−D2v )1/2

2D2v −(1+

Dv (1+



√

1−D2v )(1−D2v )1/2

√ 2 1−D2v )1/2 √1−Dv )( − n 2 2 1/2

2D2v −(1+

1−Dv )(1−Dv )

×S e(r ) Un (u, v, w ) = 0





in which case the Sheffer sequence sn (u ) n∈N0 reduces to the Chebyshev polynomials Un (u) of the second kind (see [1]). Consequently the resulting 3-variable Legendre-truncated-exponential-based Chebyshev polynomials (3VLeTEbCP) e(r ) Un (u, v, w ) of the second kind are defined by the following generating function: S

S

 2vm  4m √  ∞ (1 + m2 )r−1 n exp − J0 −u = (r ) Cn (u, v, w )m . Se 2 r r 2 2 (1 + m ) − w(−2m ) 1+m 1+m n=0

(4.1)

The other results for the 3-variable Legendre-truncated-exponential-based Chebyshev polynomials (3VLeTEbCP) (u, v, w ) of the second kind are given in Table 4. In Eq. (2.23), if we take

e(r ) Un

an,k =

⎧ 0 ⎪ ⎨

(n − k is odd )

  )k (k+1 ) ⎪ ⎩ cnc(k−1 1 + 2n (n+1 )



n [(n − k )/2]



where

1

−1

cn =



n

and

−n k

(4.2)

(n − k is even ),



 = (−1 )

k



n+k−1 , k



in which case the determinant form of sn (u ) n∈N0 reduces to the determinant form of Un (u) [23]). Consequently we find that the 3VLeTEbCP of the second kind e(r ) Un (u, v, w ) are defined by the following determinant form: S

S

e (r )

U0 (u, v, w ) = 1,

(4.3)



1



1





0



n (n+3 ) n (n+1 )

2 2 0 (r ) Un (u, v, w ) = (−1 ) 2 Se





·

·

·



0

(r ) S e1

(u, v, w )

(r ) S e2

(u, v, w )

(r ) S en−1

···

(u, v, w )

0

1 4

···

− 12

0

···

0

1 4

···

0

· · ·

· · ·

··· ··· ···

· · ·

0

0

···

 −1 n−1

0

2







n !   

2n n2 ! 1+ n2 !





0



 n(n−1)...3n!  . n+2

2n n−2 ! n 1+ ! 2 2



·



·



·



0 (r ) S en

(u, v, w )

(4.4)

Example 2. In the left-hand side of the generating function (2.16), if we take

l ( m ) = 1,





h(m ) = log(1 + m )

and

h¯ (m ) = em − 1,

in which case the Sheffer sequence sn (u ) n∈N0 reduces to the exponential polynomials φ n (u) (see [16]), we find that the resulting 3-variable Legendre-truncated-exponential-based exponential polynomials (3VLeTEbEP) e(r ) φn (u, v, w ) are defined S by the following generating function:



1 1 − w ( em − 1 )r



ev ( e

m

−1 )

∞  √ mn J0 (2(em − 1 ) −u ) = . (r ) φn (u, v, w ) Se n!

(4.5)

n=0

Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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H.M. Srivastava, M. Riyasat and S. Khan et al. / Applied Mathematics and Computation xxx (xxxx) xxx Table 5 Results for

Se

φn (u, v, w ).

(r )

S.No.

Results

I.

Series representation

II.

Multiplicative and derivative operators Differential equation

III.

Table 6 Results for

Se

(r )

Expressions n l    φn (u, v, w ) = S(n, l )S el(r ) (u, v, w ); S(n, l ) = l!1 (−1 )l− j lj jn j=0 l=0    r−1 +SeE = v + 2D−1 ( 1 + Dv ) u Dv + rwDw wDv +  ( 1 + Dv ) SeE = log   r−1 v + 2D−1 log(1 + Dv )(1 + Dv ) − n S e(r ) φn (u, v, w ) = 0 u Dv + rDw wDv Se

En(α ) (u, v, w ).

S.No.

Results

I.

Series representation

II.

(r )

Expressions Se

En(α ) (u, v, w ) =

n    n El(α ) S en(r−l) (u, v, w ) l

l=0

e v r−1 +SeE = v + 2D−1 u Dv + rwDw wDw − α eDv +1 −  = Dv SeE   (α ) eDv 2 r vDv + 2D−1 u Dv + rwDw wDv − α Dv eDv +1 − n S e(r ) En (u, v, w ) = 0

Multiplicative and derivative operators Differential equation

III.

(r )

D

The other results for the 3VLeTEbEP e(r ) φn (u, v, w ) are given in Table 5. S Now, we consider the case when an,k = s(n, k ), where s(n, k) denotes the Stirling numbers of the first kind, which arise especially in combinatorics in the study of permutations. These are linked with the Pochhammer symbol (or the shifted factorial) by the relation

(z )n := z(z + 1 ) . . . (z + n − 1 ) = 

n 

(−1 )n+k s(n, k )zk .

(4.6)

k=0



In this case the determinant form of the sequence sn (u ) n∈N0 with an,0 = δn,0 reduces to the determinant form of the exponential polynomials φ n (u) (see [23]). Consequently, for an,k = s(n, k ), we then find that the 3VLeTEbEP e(r ) φn (u, v, w ) S are defined by the following determinant form: Se

(r )

φ0 (u, v, w ) = 1,

(4.7)

(r )

S e1 (u, v, w )





s ( 1, 1 )



0 n+1

φ ( u, v , w ) = ( −1 ) ( r )

n Se

·



·



·



0

(r ) S e2

(u, v, w )

···

(r ) S en−1

(u, v, w )

s ( 2, 1 )

···

s ( n − 1, 1 )

s ( 2, 2 )

···

s ( n − 1, 2 )

· · ·

··· ··· ···

· · ·

0

···

s ( n − 1, n − 1 )

(u, v, w )



s(n, 1 )





s(n, 2 )

.

·



·



·



s(n, n − 1 )

(r ) S en

(4.8)

Example 3. If, in the left-hand side of generating function (2.16), we set



l (m ) =



in which case the Sheffer sequence sn (u )

em + 1 2



n∈N0

α

,

h (m ) = m

and

h¯ (m ) = m,

reduces to the Euler polynomials En(α ) (u ) of order α (see [16]), we find that

the resulting 3-variable Legendre-truncated-exponential-based Euler polynomials (3VLeTEbEP) are defined by the following generating function:

 em

α 2 +1



1 1 − wmr

The other results for the 3VLeTEbEP of

Se

(α )

( r ) En



Se

∞  √ mn (α ) evm J0 (2m −u ) = (u, v, w ) . ( r ) En Se n!

(α )

( r ) En

(u, v, w ) of order α )

(4.9)

n=0

(u, v, w ) of order α are given in Table 6.

Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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13



We consider the case, when

n k

an,k = where

lk =

k 

ln−k ,

(α ) j 2− j S(k, j )

j=0

and the Stirling numbers of the second kind S(n, k) are given by k 1  S(n, k ) = (−1 )k− j k!



j=0



In this case the determinant form of the sequence sn (u )



(α )

k n j . j

(4.10)

reduces to the determinant form of the Euler polynomials

n∈N0

En (u ) of order α (see [23]). Consequently, for this value of an,k , we find that the 3VLeTEbEP are defined by the following determinant form: Se

(r )

Se

(α )

( r ) En

(u, v, w ) of order α

E0(α ) (u, v, w ) = 1,

(4.11)



1





1







0



(α ) n

( u, v , w ) = ( −1 ) ( r ) En

e S

0





·

·

·



0 For α = 1, the 3VLeTEbEP (3VLeTEbEP) e(r ) En (u, v, w ).

Se

(α )

( r ) En

(r ) S e1

(r ) S e2

(u, v, w )

(u, v, w )

···

α 2

α2 + α 4 4

···

1

α

···

0

1

···

· · ·

· · ·

··· ··· ···

0

0

···







n 

−j (α ) j 2 S(n, j )

j=0



−1 n n

−j

( α ) 2 S ( n − 1 , j ) j 1

j=0



. −2 n n

−j ( α ) 2 S ( n − 2 , j )

j 2

j=0



·



·

·



nα

(r ) S en

(u, v, w )

n−1

(4.12)

2

(u, v, w ) reduce to 3-variable Legendre-truncated-exponential-based Euler polynomials

S

The algebraic approach (determinant forms) is simpler then the classical analytic approaches based on generating function methods. The determinant forms for the polynomials e(r ) Un (u, v, w ), e(r ) φn (u, v, w ) and e(r ) En(α ) (u, v, w ) provides a S S S unifying theory for these polynomials as well as for their very natural generalizations. The coefficients or the value in a chosen point can be obtained for these sequences through an efficient and stable Gaussian algorithm and may also allows the solution of the several general linear interpolation problems. 5. Graphical representations and computation of zeros In the previous section, we have considered the 3-variable Legendre-truncated-exponential-based Euler polynomials (3VLeTEbEP) e(r ) En(α ) (u, v, w ) (of order α ). Here, in this section, we present the graphical and computational aspects reS lated to these polynomials and their special cases. The explicit series representation for the 3VLeTEbEP e(r ) En (u, v, w ) is given by S

(r ) En (u, v, w ) = n! Se

[ 2n ]

n−2k

k=0

l=0

r ]  [

wl uk En−2k−rl (v ) . k! (k + 1 ) (n − 2k − rl )!

(5.1)

A 3D surface plot displays a 3-dimensional view of the surface defined by a function of two variables such that z = f (u, v ). The3D surface plots are more informative and better for analysis. The predictor variables are displayed on the u- and vscales, while the response variable w is represented by a smooth surface (3D surface plot) or a grid. Whereas on the other hand contour plots display the 3-dimensional relationship in two dimensions, with u- and v-factors (predictors) plotted on the u- and v-scales and response values represented by contours. By this we can compute the polynomial values according Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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Fig. 1.

Fig. 2.

to define arrays. A contour plot is like a topographical map in which u-, v-, and w-values are plotted instead of longitude, latitude and elevation. A contour plot provides a 2-dimensional view of the surface where points that have the same response are connected to produce contour lines of constant responses. Contour plots are useful for establishing the response values and operating required conditions. The peaks and valleys correspond with combinations of u and v that produce local maxima or minima. Both contour and 3D surface plots help to visualize the response surface. Like the contour plots, 3D surface plots are useful for establishing the derived response values and operating conditions. The 3D surface plots can provide a more clear concept of the response surface than contour plots. The software “Mathematica” is used to show the behavior of the 3D polynomials, the 3VLeTEbEP e(r ) En (u, v, w ) by means S of surface and contour plots for the special values of indices and parameters. For the even values n = 12 and r = 14, the following surface and contour plots are drawn: For the odd values n = 15 and r = 17, the following surface and contour plots are drawn: The 3D plots of a set of data points are displayed and contour plot under a surface plot of peak function are also shown (see Figs. 1–4). The graphs show the height along w-axis, which is a single valued function over a geometrically rectangular grid and specifies the value of the polynomials. The peaks and valleys correspond with combinations of u and v that produce local maxima or minima. Recently, the computing environment is making more and more rapid progress. The manual computation of these zeros is too complicated, therefore, we use “Matlab software” to investigate these zeros. By using numerical investigations and computer experiments, we find the real and complex zeros and observe the phenomenon of distribution of the zeros of Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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15

Fig. 3.

6

4

2

0

−2

−4

−6

−6

−4

−2

0

2

4

6

Fig. 4.

certain special polynomials related to the 3VLeTEbEP for specific values of index n. The investigation in this direction will lead to a new approach of employing numerical methods in the field of the hybrid special polynomials. • Taking w = 0 in 3VLeTEbEP, we obtain the 2-variable Legendre–Appell sequences (2VLeAS) [12], which by choosing   2 Pn (v ) = Sn − 1−4v , v reduce to the classical Legendre-Appell sequences P En (v ) given by following explicit series representation: n

P En ( v ) =

[2]  (−1 )k (2n − 2k )!En−2k (v ) . 2 2 k k ! ( n − k )! ( n − 2 k )!

(5.2)

k=0

• Taking u = 0 in 3VLeTEbEP, we obtain the 2-variable truncated-exponential-Appell sequences (2VTEAS) [13], which by choosing w = 1 gives the classical truncated-exponential-Euler sequences [2]e En (v ) given by following explicit series representation: n

[2]e En (v ) =

[2]  En−2k (v ) . ( n − 2 k )!

(5.3)

k=0

The real and complex zeros of P En (v ) and [2]e En (v ) are computed using “Matlab software” and are given in Tables 7 and 8. In order to make the above discussions more clear, we draw the graphs showing shapes with scattered zeros of the sequences P En (v ) and [2]e En (v ) for odd and even indices. Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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H.M. Srivastava, M. Riyasat and S. Khan et al. / Applied Mathematics and Computation xxx (xxxx) xxx Table 7 Real zeros of P En (v ) and

Table 8 Complex zeros of P En (v ) and

[2]e En

[2]e En

( v ).

Degree n

P En ( v )

[2]e En

1 2 3 4 5 6 7 8 9 10

0.5000 1.0401, −0.0401 1.4298, 0.3908, −0.3206 1.8597, 0.7658, −0.6194, −0.0061 1.8294, 1.5052, 0.50 0 0, −0.8294, −0.5252 1.0 0 01, −0.0 0 01 1.4999, 0.50 0 0, −0.4999 2.6515, 0.0000 5.3184, −1.6048, 1.7156, 0.4220, −0.4220 −2.7139, 0.6936, −0.6936, −0.0 0 0 0

0.5000 – 0.5159 – 0.5000 – 0.4987 – 0.5000 –

( v ).

(v )

Degree n

P En

1 2 3 4

– – – –

5

–

6

1.9117 + 0.2694i, 1.9117 − 0.2694i −0.9117 + 0.2694i, −0.9117 − 0.2694i 2.0869 + 0.4371i, 2.0869 − 0.4371i −1.0869 + 0.4371i, −1.0869 − 0.4371i 2.1538 + 0.9848i, 2.1538 − 0.9848i, −1.4770 + 0.6500i −1.4770 − 0.650 0i, −0.0 026 + 0.6906i, −0.0026 − 0.6906i −0.4738 + 2.3540i, −0.4738 − 2.3540i 0.0092 + 1.0531i, 0.0092 − 1.0531i 3.9298 + 1.6595i, 3.9298 − 1.6595i, 0.5782 + 1.5258i 0.5782 − 1.5258i, −0.6511 + 1.4804i, −0.6511 − 1.4804i

7 8 9 10

[2]e En

Se

S.No.

Results

I.

Operational rule

II. III.

(r )

Bn,μ (u, v, w; β ) and 3VLeTEbES

Generating function Series Expansion

3VLeTEbBS



β

Se

(r )

(r )

(r )

[2]e En

(v ) give the numerical results for the approximate solutions of the

En,μ (u, v, w; β ).

Bn,μ (u, v, w; β )

 ∂ r −μ { B (u, v )} = − w ∂∂w w ∂v (r ) Bn,μ (u, v, w; β ) S n r Se

√ (e p −1 )ev p J0 (2 p −u ) p(β −pr wDw w )μ

Se

Se

(v )

– 0.50 0 0 + 1.3229i, 0.50 0 0 − 1.3229i 0.4921 + 2.2913i, 0.4921 − 2.2913i 0.4672 + 2.7908i, 0.4672 − 2.7908i 0.5328 + 1.6473i, 0.5328 − 1.6473i 1.3868 + 3.0881i, 1.3868 − 3.0881i −0.3868 + 3.0881i, −0.3868 − 3.0881i 1.9609 + 3.7469i, 1.9609 − 3.7469i, −0.9609 + 3.7469i −0.9609 − 3.7469i, 0.50 0 0 + 1.5623i, 0.50 0 0 − 1.5623i 2.5494 + 4.2508i, 2.5494 − 4.2508i, −1.5497 + 4.2511i −1.5497 − 4.2511i, 0.5010 + 3.0017i, 0.5010 − 3.0017i 3.2061 + 4.7319i, 3.2061 − 4.7319i, −2.2061 + 4.7319i, −2.2061 − 4.7319i 0.50 0 0 + 4.0490i, 0.50 0 0 − 4.0490i, 0.50 0 0 + 1.5717i, 0.50 0 0 − 1.5717i 3.8743 + 5.1981i, 3.8743 − 5.1981i, −2.8743 + 5.1981i, −2.8743 − 5.1981i 0.50 0 0 + 4.6540i, 0.50 0 0 − 4.6540i, 0.50 0 0 + 3.1738i, 0.50 0 0 − 3.1738i 4.5588 + 5.6380i, 4.5588 − 5.6380i, −3.5588 + 5.6380i, −3.5588 − 5.6380i 1.1469 + 4.8215i, 1.1469 − 4.8215i, −0.1469 + 4.8215i, −0.1469 − 4.8215i 0.50 0 0 + 1.5707i, 0.50 0 0 − 1.5707i

Note. From Table 7, we note that the real zeros of the sequences P En (v ) and equations P En (v ) = 0 and [2]e En (v ) = 0 for n=1–10. Table 9 Results for 3VLeTEbBS

(v )

=

∞  n=0

Bn,μ (u, v, w; β ) np! n

Se

(r )

Bn,μ (u, v, w; β ) = n!

[ n/r] [n/ 2] s=0 l=0

Bn−2l−rs(v ) ws ul (μ )s β s (n−2l−rs )! (l! )2

3VLeTEbES



β

Se

(r )

√

(e p +1 )ev p J0 (2 p −u ) 2(β −wDw wpr )μ Se

(r )

En,μ (u, v, w; β )

 ∂ r −μ { E (u, v )} = − w ∂∂w w ∂v (r ) En,μ (u, v, w; β ) S n r Se =

∞  n=0

En,μ (u, v, w; β ) np! n

Se

(r )

En,μ (u, v, w; β ) = n!

[ n/r] [n/ 2] s=0 l=0

En−2l−rs(v ) ws ul (μ )s β s (n−2l−rs )! (l! )2

The above graphs indicate the behavior of the polynomials for odd and even indices. Since, an nth degree polynomial has at most (n − 1 ) turning points or relative extrema. Thus, the following conclusions are drawn: Figs. 5 and 6 show that the polynomials P E10 (v ) and P E5 (v ) have three and two turning points, respectively and Figs. 7 and 8 show that [2]e E10 (v ) and [2]e E5 (v ) have one and two turning points, respectively. 6. Integral transforms The schemata for applications of the integral transforms of mathematical physics to differential, integral, integrodifferential equations and in the theory of special functions has been developed. In the present section, we propose to study an appropriate combination of the operational approach with integral transforms to give a better insight to the 3-variable Legendre-truncated-exponential-based Appell sequences. Fractional calculus is a collection of relatively little-known mathematical results concerning generalizations of differentiation and integration to non-integer orders. While these results have been accumulated over centuries in various branches of mathematics, they have until recently found little appreciation or application in physics and other mathematically oriented Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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14

17

Shape and zeros of PE 10(v)

10 9

12 10

Im(v)

8 6 4 2 0 -2 -3

-2

-1

0

1

2

3

4

Re(v) Fig. 5.

2

Shape and zeros of PE 5(v)

10 4

0

Im(v)

-2

-4

-6

-8

-10

-12 -3

-2

-1

0

1

2

3

4

5

Re(v) Fig. 6.

Shape and zeros of

E (v)

[2]e 10

40 35 30 25

Im(v)

20 15 10 5 0 -5 -10 -4

-3

-2

-1

0

1

2

3

4

5

Re(v) Fig. 7.

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Shape and zeros of

E (v)

[2]e 5

15 10 5 0

Im(v)

-5 -10 -15 -20 -25 -30 -35 -4

-3

-2

-1

0

1

2

3

4

Re(v) Fig. 8.

sciences. This situation is changing and now a number of research areas in physics are growing, which employ fractional calculus. The integral transforms can be employed for derivation of the closed form solutions to some integral equations of convolution type and to the integral, differential or integro-differential equations with the generating operators. They also find applications in the Mikusinski-type operational calculi. The combined use of integral transforms and special polynomials provides a powerful tool to deal with fractional derivatives, see for example [10]. Since the differentiation and integration are usually regarded as discrete operations, therefore it is useful to evaluate a fractional derivative. The fractional derivatives arise as the infinitesimal generators of a class of translation invariant convolution semigroups, which appear universally as attractors for coarse graining procedures or scale changes. They are parameterized by a number in the unit interval corresponding to the order of the fractional derivative. We recall the following definitions: Definition 3. The Euler  -function [2] is given by



(x ) =

∞

mx−1 e−x dm,

0

Re(x ) > 0

(6.1)

Definition 4. The Euler’s integral [21, p.218] is given by (see also [10])

a −μ =



1

(μ )

∞

e−am mμ−1 dm,

0

min{Re(μ ), Re(a )} > 0.

(6.2)

The Euler’s integral form the basis of new generalizations of special polynomials. The combination of the properties of exponential operators with suitable integral representations for the special polynomials yields an efficient way of treating fractional operators, for example see [5,6,10]. Operational methods can be exploited to simplify the derivation of properties associated with ordinary and generalized special functions. Operational techniques including differential and integral operators yield a systematic and analytic approach to study special functions. In order to establish the properties and applications of the above mentioned generalized Appell sequences via fractional operators, we establish the following operational rules: We note that in view of generating Eq. (3.4), the 3VLeTEbAS are solutions of the following equation:

∂r ∂ (r ) A (u, v, w ) = (r ) A (u, v, w ) ∂vr S e n ∂ (wDw w ) S e n under the initial condition: Se

(r )

An (u, v, 0 ) = S An (u, v ).

(6.3)

(6.4)

In view of the above equations, we find the following operational representation between the 3VLeTEbAS and the 2VLeAS: Se

(r )

An (u, v, w ) = exp (wDw wDrv ){S An (u, v )}.

(6.5)

Again, from generating Eq. (3.4), we find that the 3VLeTEbAS are solutions of the following equation:

∂ ∂ ∂2 u (r ) An (u, v, w ) = (r ) A (u, v, w ), e ∂ u ∂ u Se n ∂v2 S

under the following initial condition: Se

(r )

An ( 0, v, w ) = e ( r ) An ( v, w )

(6.6) (6.7)

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From Eqs. (6.6) and (6.7), the following operational representation between the 3VLeTEbAS and 2VTEAS is obtained:



Se

(r )

An (u, v, w ) = exp

D−1 u



We also have Se

(r )

An (u, v, w ) = exp

D−1 u

∂2 { ( r ) A ( v, w )}. ∂v2 e n

(6.8)

∂2 ∂r + wDw w r {An (v )}. ∂v ∂v2

(6.9)

We desire to combine the exponential operators with integral transforms, so that the family of 3VLeTEbAS e(r ) An (u, v, w ) S evolve a new class of sequences with new parameters, we denote this, as class of the 3-variable generalized Legendretruncated-exponential-based Appell sequences (3VGLeTEbAS) by e(r ) An,μ (u, v, w; β ). S

Theorem 5. For the generalized Legendre-truncated-exponential-based Appell sequence tional rule holds true: 

β −w Proof. Replacing a by S An (u, v ),

∂ ∂r w r ∂ w ∂v

−μ

Se

(r ) An,μ

(u, v, w; β ), the following opera-

{S An (u, v )} = S e(r) An,μ (u, v, w; β ).

(6.10)

  ∂r β − w ∂∂w w ∂v in integral (6.2) and then operating the resultant equation on the 2VLeAS r

we find

 β −w

∂ ∂r w r ∂ w ∂v

−μ S An

(u, v ) =



1

(μ )

∞

 e−β m mμ−1 exp mw

0

∂ ∂r w r ∂ w ∂v

S An

(u, v )dm,

(6.11)

which on use of Eq. (6.5) gives

 −μ  ∞ ∂ ∂r 1 e−β m mμ−1 S e(r ) An (u, v, mw )dm. β −w w r S An (u, v ) = ∂ w ∂v (μ ) 0

The transform on the right hand side of Eq. (6.12) can be considered as a new class of 3VGLeTEbAS such that  ∞ Se

(r )

An,μ (u, v, w; β ) =

1

(μ )

e−β m mμ−1 S e(r ) An (u, v, mw )dm.

0

(6.12) Se

(r ) An,μ

(u, v, w; β ) (6.13)



Thus, with use of Eqs. (6.12) and (6.13), assertion (6.10) follows.

Theorem 6. For the generalized Legendre-truncated-exponential-based Appell sequences erating function holds true:

Se

(r ) An,μ

(u, v, w; β ), the following gen-

√ ∞  ev p J0 (2 p −u ) pn = . (r ) An,ν (u, v, w; α ) e S l ( p)(β − wDw wpr )μ n!

(6.14)

n=0

Proof. Multiplying both sides of Eq. (6.13) by ∞ 

(r ) An,μ Se

(u, v, w; β )

n=0

pn n!

and summing over n, we find ∞

 1 pn = n! (μ ) n=0



∞

e−β m mμ−1 S e(r ) An (u, v, mw )

0

pn dm, n!

(6.15)

which on using Eq. (3.4) in the right hand side becomes ∞ 

(r ) An,μ Se

(u, v, w; β )

n=0

  √  pn ev p J0 (2 p −u ) ∞ − β −wDw wpr m μ−1 = e m dm. n! (μ )l ( p) 0

Application of Eq. (6.2) in the right hand side of the above equation yields assertion (6.14). Theorem 7. For the generalized Legendre-truncated-exponential-based Appell sequences plicit series expansion holds true:



[n/r] [n/2] Se

(r )

An,μ (u, v, w; α ) = n!

s=0 l=0

Se

(r ) An,μ

An−2l−rs(v ) ws ul (μ )s . β s (n − 2l − rs )! (l! )2

(6.16)



(u, v, w; β ), the following ex-

(6.17)

Proof. Using the following series expansion of 3VLeTEbAS:



[n/r] [n/2] Se

(r )

An (u, v, w ) = n!

s=0 l=0

An−2l−rs(v ) ws ul , (n − 2l − rs )! (l! )2

(6.18)

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in the r.h.s of Eq. (6.13), we have

S

e (r )

An,μ (u, v, w; β ) =

(μ + s ) (μ )(μ + s )

which in view of Eq. (6.2) yields assertion (6.17). Remark 3. For β = μ = 1 and w →

D−1 w ,

exponential polynomials 3VGLeTEP are given as:

and

S

∞

e−β m mμ+s−1 n!

0



[n/r] [n/2] s=0 l=0

An−2l−rs(v ) ws ul dm, (n − 2l − rs )! (l! )2

(6.19)



the 3VGLeTEbAS

Se

(r ) An,μ

(u, v, w; β ) reduce to the 3VLeTEbAS

Se

( r ) An

(u, v, w ).

(u, v, w; β ) reduce to the 3-variable generalized Legendre-truncatedn,μ (u, v, w; β ). The operational rule and generating function for these polynomials

Remark 4. For l (m ) = 1, the 3VGLeTEbAS e (r )



Se

(r ) An,μ

 −μ ∂ ∂r {vn } = S e(r ) n,μ (u, v, w; β ) β −w w r ∂ w ∂v

(6.20)

√ ∞  evm J0 (2m −u ) mn = , (r ) An,μ (u, v, w; α ) Se r μ (β − m wDw w ) n!

(6.21)

n=0

respectively. The recurrence relations are of fundamental importance in analysis of algorithms. In digital signal processing, recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in infinite impulse response (IIR) digital filters. The linear recurrence relations are also used extensively in both theoretical and empirical economics. The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. The problems arising in different areas of science and engineering are usually expressed in terms of differential equations, which in most of the cases have special functions as their solutions. During the past three decades, the development of nonlinear analysis, dynamical systems and their applications to science and engineering has stimulated renewed enthusiasm for the theory of differential equations. Differential equations play an important role in modeling virtually every physical, technical or biological process, from celestial motion, to bridge design, to interactions between neurons. Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. Next, we find the differential recurrence relations for the 3VGLeTEAS e(r ) An,μ (u, v, w; β ). In order to achieve this, we S differentiate generating function (6.14) with respect to v, w and β , so that we have

∂ { (r) A (u, v, w; β )} = n ∂v S e n,μ

Se

(r )

An−1,μ (u, v, w; β ),

∂ { (r) A (u, v, w; β )} = μ n(n − 1 ) . . . (n − r + 1 ) ∂ w S e n,μ ∂ { (r) A (u, v, w; β )} = −μ ∂β S e n,μ

Se

(r )

Se

(r )

(6.22)

An−r,μ+1 (u, v, w; β ),

An,μ+1 (u, v, w; β ).

(6.23)

(6.24)

It happens very often that the solution of a given problem in physics or applied mathematics requires the evaluation of infinite sums involving special polynomials. There is a continuous demand of solving problems by means of formulas and identities in research fields like classical and quantum optics. These formulas, functional equations and identities arise in well-defined combinatorial contexts and they lead systematically to well defined classes of functions. The summation formula of hybrid type special polynomials of more than one variable often appear in applications ranging from electromagnetic processes to combinatorics.

(u, v, w; β ), the following explicit summation formula in terms of the generalized Legendre-truncated-exponential polynomials S e(r ) n,μ (u, v, w; β ) and Appell Theorem 8. For the generalized Legendre-truncated-exponential-based Appell sequences polynomials An (y) holds true:

S

e (r )

 

An,μ (u, v, w; β ) =

n  n  n l l=0 s=0

Se

(r ) An,μ



n−l (−y )l As (y )S e(r ) n−l−s,μ (u, v, w; β ). s

(6.25)

Proof. Consider the product of generating functions (1.1) and (2.5) in the following form:

√ ∞  ∞  eym J0 (2m −u )evm mn+s = As (y )S e(r ) n,μ (u, v, w; β ) . r μ l (m )(β − m wDw w ) n!s!

(6.26)

n=0 s=0

Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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21

which on simplification becomes

  √ ∞  n  n  J0 (2m −u )evm mn n n−l l (r ) = ( −y ) A ( y ) e ( u, v , w ; β ) . s n −l−s, μ S l s l (m )(β − mr wDw w )μ n!

(6.27)

n=0 l=0 s=0

Finally, using generating function (6.14) in the l.h.s. of Eq. (6.27) and then equating the coefficients of like powers of m in the resultant equation, assertion (6.25) follows.  By making suitable selections for the function l(p) in the results derived above of the 3VGLeTEbAS S e(r ) n,μ (u, v, w; β ), we get the corresponding results for the particulars members belonging to this class. (i) Taking l ( p) = e pp−1 in the 3VGLeTEbAS e(r ) An,μ (u, v, w; β ), we get the 3-variable generalized Legendre-truncatedS exponential-based Bernoulli sequence (3VLeTEbBS) e(r ) Bn,μ (u, v, w; β ). S

(ii) Similarly, taking l ( p) = e p2+1 in the 3VGLeTEbAS e(r ) An,μ (u, v, w; β ), we get the 3-variable generalized LegendreS truncated-exponential-based Euler sequence (3VLeTEbES) e(r ) En,μ (u, v, w; β ). S The corresponding results for the 3VLeTEbBS and 3VLeTEbES are given in Table 8. Several identities involving Bernoulli and Euler polynomials are known in literature. The operational formalism can be utilized to derive the corresponding identities involving generalized Legendre-truncated-exponential-based Bernoulli and Euler polynomials. To achieve this, we establish the following operational rule: Theorem 9. The following operational rule between the generalized Legendre-truncated-exponential-based Appell sequences e(r ) An,μ (u, v, w; β ) and Appell sequences An (v ) holds true: S



exp D−1 u

∂2 ∂v2

 −μ ∂r β − wDw w r {An (v )} = S e(r) An,μ (u, v, w; β ). ∂v

(6.28)

Proof. We consider the following operation: ( ): Replacement of w by wm, multiplication by (1ν ) e−β m mμ−1 and then integration with respect to m from m = 0 to m = ∞. Now, operating ( ) on both sides of equation (6.9) and then using the Weyl identity [8, p.7] in the l.h.s. and equation (6.13) in r.h.s. of the resultant equation, we find

  ∞ − β −wD w ∂ r m 2 w r ∂ 1 ∂v exp D−1 e mμ−1 An (v )dm = S e(r ) An,μ (u, v, w; β ), u ∂v2 (μ ) 0 

which on using equation (6.2), yields assertion (6.28).

(6.29)



In order to give applications rule (6.28), we use the following operation:   of the operational r −μ −1 ∂ 2 ∂ O: Operating exp Du ∂v2 β − wDw w ∂vr on both sides of a given result.

Consider the following functional equations involving Bernoulli and Euler polynomials Bn (v ) and En (v )[14]:

Bn ( v + 1 ) − Bn ( v ) = n n−1  m=0

vn−1 ,

n = 0, 1, 2 . . . ,

(6.30)



n Bm (v ) = nvn−1 , m

n = 2, 3, 4 . . . .

(6.31)

En ( v + 1 ) + En ( v ) = 2vn ,

En ( mv ) = mn

m −1  k=0

(6.32)

 k (−1 )k En v + n = 0, 1, 2 . . . ; m odd. m

(6.33)

Performing operation (O ) on both sides of above equations and then using operational rule (6.28) (corresponding to the Bernoulli and Euler polynomials) and relation (6.20), we obtain the following identities involving generalized Legendretruncated-exponential-based Bernoulli e(r ) Bn,μ (u, v, w; β ) and Euler polynomials e(r ) En,μ (u, v, w; β ): S

S

(r ) (r ) Bn,μ (u, v + 1, w; β ) − e(r ) Bn,μ (u, v, w; β ) = n S e n−1,μ (u, v, w; β ), Se S n−1 

n = 0, 1, 2 . . . .

(6.34)



m=0 Se

n m

(r )

Se

(r )

Bm,μ (u, v, w; β ) = nS e(r ) n−1,μ (u, v, w; β ),

n = 2, 3, 4 . . . .

En,μ (u, v + 1, w; β ) + S e(r ) En,μ (u, v, w; β ) = 2S e(r ) n,μ (u, v, w; β ),

(6.35) (6.36)

Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683

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Se

(r )

En,μ (u, mv, w; β ) = mn

m −1  k=0

 k (−1 )k S e(r) En,μ u, v + , w; β n = 0, 1, 2 . . . ; m odd. m

(6.37)

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Please cite this article as: H.M. Srivastava, M. Riyasat and S. Khan et al., A new approach to Legendre-truncatedexponential-based Sheffer sequences via Riordan arrays , Applied Mathematics and Computation, https://doi.org/10.1016/j. amc.2019.124683