Numerical Computation of Zeros of Certain Hybrid q-Special Sequences

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Procedia Computer Science 00 (2019) 000–000 Procedia Computer Science (2019) 000–000 Procedia Computer Science 15200 (2019) 166–171 Procedia Computer Science 00 (2019) 000–000

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International Conference on Pervasive Computing Advances and Applications - PerCAA 2019 www.elsevier.com/locate/procedia International Conference on Pervasive Computing Advances andof Applications Numerical Computation of Zeros Certain - PerCAA 2019 International Conference on Pervasive Computing Advances and Applications - PerCAA 2019

Hybrid q-Specialof Sequences Numerical Computation Zeros of Certain Numerical Computation of Zeros of Certain Subuhi Khan∗, Tabinda Nahid Hybrid q-Special Sequences Hybrid q-Special Sequences Department of Mathematics, Aligarh Muslim University, Aligarh, India Subuhi Khan∗, Tabinda Nahid Subuhi Khan∗, Tabinda Nahid

Department of Mathematics, Aligarh Muslim University, Aligarh, India

Abstract

Department of Mathematics, Aligarh Muslim University, Aligarh, India

This article is written with an aim to construct a new family of the q-Bessel-Appell sequences, which may find applications in Abstract physics and biological sciences. The investigation includes derivation of generating equation, series expansion and determinant Abstract form article for theisq-Bessel-Appell The corresponding results are q-Bessel-Appell illustrated for particular members of this family. This written with an family. aim to construct a new family of the sequences, which may findq-hybrid applications in Further, the graphical representation for these sequences are shown and their zeros are computed. physics and biological sciences. The investigation includes derivation of generating equation, series expansion and determinant This article is written with an aim to construct a new family of the q-Bessel-Appell sequences, which may find applications in form forand thebiological q-Bessel-Appell family. The corresponding are illustrated for particular of this q-hybrid family. physics sciences. The investigation includesresults derivation of generating equation, members series expansion and determinant c 2019  Authors. Publishedfamily. by Elsevier Ltd. Further, the representation forThe these sequences areresults shownare andillustrated their zerosforareparticular computed. form forThe thegraphical q-Bessel-Appell corresponding members of this q-hybrid family. This is antheopen accessrepresentation article under the BY-NC-ND (https://creativecommons.org/licenses/by-nc-nd/4.0/). Further, graphical forCC these sequenceslicense are shown and their zeros are computed. © 2019 The Authors. Published by Elsevier Ltd. c 2019 The Authors. Published by Elsevier Ltd.  This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Keywords: q-Appell sequences; q-Bessel sequences; q-Bessel-Appell sequences. This is an open access article under the BY-NC-ND license c 2019  The Authors. Published by Ltd. committee Peer-review under responsibility ofElsevier theCC scientific of(https://creativecommons.org/licenses/by-nc-nd/4.0/). the International Conference on Pervasive Computing Advances This is an open access article2019. under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/). and Applications – PerCAA Keywords: q-Appell sequences; q-Bessel sequences; q-Bessel-Appell sequences. Keywords: q-Appell sequences; q-Bessel sequences; q-Bessel-Appell sequences.

1. Introduction and preliminaries 1. Recently, Introduction preliminaries a lotand of research has been done in computing environment using many software. The investigation in this field is very beneficial for researchers to comprehend the concepts of q-sequences and q-numbers. Use of computer 1. Introduction and preliminaries Recently, a lot ofusresearch has been in computing environment software. The investigation in this softwares enables to understand thedone concepts very easily. In recentusing years,many the computer software is extended to field is very beneficial for researchers to comprehend the concepts of q-sequences and q-numbers. Use of computer solve problems arising in mathematics. Recently, a lot of research has been done in computing environment using many software. The investigation in this softwares enables us tofor understand the concepts very the easily. In recent years, the computer software is of extended to field is very beneficial researchers comprehend concepts of q-sequences andRiyasat q-numbers. Use computer The Bessel polynomials p s (u) have toemerged in numerous branches of science. M. and Subuhi Khan [7] solve problems arising in mathematics. softwares us to understand the concepts very peasily. In recent years, the computer software is extended to constructedenables the q-extension of the Bessel polynomials s (u) and studied certain properties of these q-special polynosolve problems arising in mathematics. TheInBessel p (u) have emerged inpnumerous branches of the science. M. Riyasat Subuhi Khan [7] (u), the structure q-extension of theand Bessel polynomials mials. order polynomials to study the q-Bessel polynomials s

s,q

studiedofcertain properties of these polynoconstructed thepolynomials q-extension the of Bessel polynomials prelated p s (u) required. A realistic of study q-Bessel andinother polynomials will provide a new perspective utilize Theis Bessel p s (u) have emerged numerous branches science. M. Riyasat and q-special Subuhithat Khan [7] s (u) and theand structure the q-extension Bessel polynomials mials. In order study the of q-Bessel polynomials p s,q (u), numerical methods. studiedofcertain properties of of the these q-special polynoconstructed the to q-extension the Bessel polynomials p s (u) p s (u) isInrequired. realistic study of q-Bessel and other relatedstructure polynomials will provide a new perspective that utilize of the q-extension the Bessel polynomials mials. order this toAstudy q-Bessel polynomials p s,q (u), Throughout paperthestandard notations are used. Thethe following q-standard notationsofand definitions are taken numerical methods. p (u) is required. A realistic study of q-Bessel and other related polynomials will provide a new perspective that utilize s from [2]: numerical methods. Throughout this paper standard notations are used. The following q-standard notations and definitions are taken from [2]: Throughout this paper standard notations are used. The following q-standard notations and definitions are taken ∗ Corresponding author. Tel.: +0- +0-9412878837. from [2]: E-mail address: [email protected]

∗

Corresponding author. Tel.: +0- +0-9412878837. c 2019 The Authors. Published by Elsevier Ltd. 1877-0509  address: [email protected] ∗ E-mail author. Tel.: +0- the +0-9412878837. ThisCorresponding is an open access article under CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/). E-mail address: [email protected] 1877-0509 © 2019 The Authors. Published Elsevier Ltd. Ltd. c 2019 The Authors. Published by 1877-0509  by Elsevier This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/). c 2019 1877-0509  Authors. Published Elsevier Ltd. Peer-review underThe responsibility of thebyscientific committee of the International Conference on Pervasive Computing Advances This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/). and Applications – PerCAA 2019. 10.1016/j.procs.2019.05.039

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Subuhi Khan et al. / Procedia Computer Science 152 (2019) 166–171 Author name / Procedia Computer Science 00 (2019) 000–000

167

• q-analogue of ν ∈ C is specified by: [ν]q =

1 − qν , q ∈ C\{1}. 1−q

(1)

• q-factorial function is specified by: [µ]q ! =

µ  s=1

[s]q , µ ∈ N, [0]q ! = 1, q ∈ (0, 1) ∈ C.

• q-analogue of binomial coefficient is specified by:   [λ]q ! λ , δ = 0, 1, . . . , λ, λ ∈ N0 . = δ q [δ]q ![λ − δ]q !

(2)

(3)

• q-exponential function is defined as: eq (u) =

∞  ul , |q| ∈ (0, 1). [l]q ! l=0

(4)

In the last few decades, the engrossment in the q-Appell sequences in several fields have moderately incremented, see for example [1, 6]. The generating equation for q-Appell polynomials A s,q (u) is given by [1]: Aq (w)eq (uw) =

∞ 

A s,q (u)

s=0

ws , [s]q !

(5)

where Aq (w) :=

∞  s=0

A s,q

ws , [s]q !

Aq (w)  0

(6)

is analytic at w = 0. It is to be noted that for q = 1, A s,q (u) reduce to the Appell polynomials A s (u) [3]. The generating equation and series expansion for q-Bessel polynomials p s,q (u) are as follows [7]: eq (u(1 −

√

1 − 2w)) =

∞ 

p s,q (u)

s=0

ws [s]q !

(7)

and p s,q (u) =

s−1  [s − 1 + i]q ! u s−i , [s − 1 − i]q ! [i]q ! 2i i=0

(8)

respectively. The hybrid type q-special polynomials are topic of recent interest. In this article, the q-Bessel-Appell polynomial sequences are introduced. The series representation and determinant representation for the q-Bessel-Appell polynomials are investigated. Further, the corresponding results for q-Bessel based Bernoulli and Euler polynomials are obtained. The graphical representation of some q-special polynomials are drawn and their zeros are also explored. In addition, nature of these zeros are displayed using computer software. 2. q-Bessel-Appell polynomials First, generating equation of q-Bessel-Appell polynomials (denoted by p A s,q (u)) is obtained as:

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168

3

Result 2.1 For q-Bessel-Appell polynomials p A s,q (u), the following generating equation holds true: Aq (w)eq (u(1 −

√

1 − 2w)) =

∞ 

p A s,q (u)

s=0

ws . [s]q !

(9)

Proof. Using expansion (4) in equation (5) and then changing the powers of u0 , u1 , u2 , ..., u s of u with correlating expressions p0,q (u), p1,q (u), ...., p s,q (u) and summing up the terms in obtained relation and indicating resultant q-Bessel-Appell polynomials by p A s,q (u), relation (9) is proved. Result 2.2 The series expansion for q-Bessel-Appell polynomials p A s,q (u) is:

p A s,q (u) =

s    s i=0

i

Ai,q p s−i,q (u).

(10)

q

Proof. Utilizing equations (6) and (7) in generating relation (9) and then arranging series, we find s   ∞  ∞   ws ws s = . Ai,q p s−i,q (u) p A s,q (u) [s]q ! s=0 [s]q ! i q s=0 i=0

(11)

comparison of identical powers of w on the foregoing relation, verify relation (10). Because of the significance of determinant representation for applied and computing purposes, the determinant definition for p A s,q (u) is obtained utilizing a same method [6, p. 359 (Theorem 7)] and taking help of relations (5) and (7). Result 2.3 The determinant representation of q-Bessel-Appell polynomials p A s,q (u) of degree s is: p A0,q (u)

p A s,q (u)

=

=

1 β0,q ,

s

(−1) (β0,q ) s+1

  1 p1,q (u) p2,q (u)   β2,q  βo,q β1,q  2   0 β0,q 1 q β1,q    0 0 β0,q  .. ..  .. . .  .  0 0 0

· · · p s−1,q (u) ··· ··· ··· .. . ···

β s−1,q  s−1 1

q

 s−1 2

β s−2,q

β s−3,q .. .

q

β0,q

 p s,q (u)    β s,q     s  β s−1,q 1 q  ,   s  2 q β s−2,q   ..  .  s  s−1 q β1,q 

(12)

where s = 1, 2, ...; p s,q (u) (s = 0, 1, 2, ....):= q-Bessel polynomials; β0,q  0; β0,q = β s,q =

1 A0,q ,

 s    s

 − A10,q 

i q Ai,q i=1

  β s−i,q  ,

s = 1, 2, ...

.

(13)

In the forthcoming section, certain examples are constructed to give the application of the results established here. 3. Examples The q-analogues of Bernoulli and Euler polynomials are important members of q-Appell family. These polynomials perform indispensable applications in several formulae that are very beneficial in classical and numerical analysis.

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169

w Example 3.1 In consideration of Aq (w) = eq (w)−1 , A s,q (u) reduce to q-Bernoulli polynomials Bs,q (u) [4]. Accordingly, w 1 (r = 1, 2, · · · , s) in determinant definition taking Aq (w) = eq (w)−1 in equations (9), (10) and β0,q = 1 and βr,q = [r+1] q (12), we obtain the following generating function, series expansion and determinant form for p Bs,q (u):= q-BesselBernoulli polynomials: ∞  √ ws w eq (u(1 − 1 − 2w)) = , (14) p B s,q (u) eq (w) − 1 [s]q ! s=0 s    s Bi,q p s−i,q (u) (15) p B s,q (u) = i q i=0

and

p B0,q (u)

= 1,

  1 p1,q (u) p2,q (u)   1 1  1 [2]q [3]q   2  1 1 1 q [2]q s  0 p B s,q (u) = (−1)    1  0 0  . . .. ..  .. .  0 0 0

respectively.

· · · p s−1,q (u) 1 [s]q

··· ··· ··· .. . ···

 s−1 1

1 q [s−1]q

 s−1 2

1 q [s−2]q

.. . 1

 p s,q (u)    1  [s+1]q    s 1  1 q [s]q   , s = 1, 2, · · · ,   s 1  2 q [s−1]q   ..   s .  1  s−1 q [2]q 

(16)

2 Example 3.2 In consideration of Aq (w) = eq (w)+1 , A s,q (u) reduce to q-Euler polynomials E s,q (u) [4]. Accordingly, 2 taking Aq (w) = eq (w)+1 in equations (9), (10) and β0,q = 1 and βr,q = 12 (r = 1, 2, · · · , s) in determinant definition (12), we obtain the following generating function, series expansion and determinant form for p E s,q (u):= q-Bessel-Euler polynomials: ∞  √ ws 2 eq (u(1 − 1 − 2w)) = , (17) p E s,q (u) eq (w) + 1 [s]q ! s=0 s    s Ei,q p s−i,q (u) (18) p E s,q (u) = i q i=0

and

p E 0,q (u)

= 1,

   1 p1,q (u) p2,q (u) · · · p s−1,q (u) p s,q (u)      1 1 1 1 1 · · ·   2 2 2 2          2 s−1 s 1 1 1  0 1 1 q 2 ··· 1 q 2 1 q 2  s   , s = 1, 2, · · · , p E s,q (u) = (−1)     s−1  s  1 · · · 2 12 2 12   0 0 q q   . .. .. .. .. ..   .. . . . . .    s 1  0 ··· 1 0 0 s−1 q 2 

(19)

respectively.

The forthcoming section aims to plot the shapes of q-Bessel polynomials p A s,q (u) and related members belonging to this family and their zeros are also explored.

Subuhi Khan et al. / Procedia Computer Science 152 (2019) 166–171

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Author name / Procedia Computer Science 00 (2019) 000–000

Fig. 1. Shape of p s,1/2 (u);

5

Shape of p Bs,1/2 (u);

Shape of p E s,1/2 (u)

4. Graphical representations and computation of zeros The purpose of this section is to demonstrate the advantage of using numerical investigation to find fascinating new pattern of zeros of the q-Bessel polynomials p s,q (u) and certain members belonging to the q-Bessel-Appell family p A s,q (u). This will provide a new perspective for comparison between the parent q-Bessel polynomials and corresponding hybrid members of q-Bessel-Appell class. First, the graphs of p s,q (u), p Bs,q (u) and p E s,q (u) have been plotted for s = 1, 2, 3, 4 and q = 12 (0 < q < 1). Using the values of corresponding numbers Bs,q and E s,q [4] and in view of definitions (8), (15) and (18), expressions of p s,q (u), p B s,q (u) and p E s,q (u) are obtained for s = 0, 1, 2, 3, 4. In Table 1, first five expressions of these polynomials are listed. Table 1. Expressions of first five p s,q (u), p Bs,q (u) and p E s,q (u) s

0

p s,1/2 (u) p B s,1/2 (u) p E s,1/2 (u)

1 1 1

1

2

u u− u−

u2

+ − u2 −

2 3 1 2

u2

3 3 4u 1 2 4 u + 21 1 8

u3

+ − u3 + u3

4 21 2 16 u 7 2 48 u 7 2 16 u

+ + −

105 128 u 172 1 1536 u + 45 11 3 u + 128 64

u4 + u4 + u4 +

105 3 3255 2 29295 64 u + 2048 u + 32768 u 25 3 965 2 1207 64 u + 6144 u − 1536 u + 45 3 2345 2 1635 64 u − 2048 u + 2048 u +

6594583 7110656 31311 32768

With the help of Matlab and by using the expressions of the p s,q (u), p Bs,q (u) and p E s,q (u) from Table 1 for q = and s = 1, 2, 3, 4, Figure 1 is drawn.

1 2

Next, we investigate the zeros of polynomials mentioned in Table 1. The zeros of p s,1/2 (u), p Bs,1/2 (u) and p E s,1/2 (u) are computed by using Matlab and are listed in Tables 2 and 3. Table 2. Real zeros of p s,1/2 (u), p Bs,1/2 (u) and p E s,1/2 (u) Degree s

p s,1/2 (u)

p B s,1/2 (u)

p E s,1/2 (u)

1 2 3 4

0 0, −0.75 0 0, -0.9562

0.666667 complex zeros −0.144403 complex zeros

0.5 0.353553, −0.353553 −0.670124 −1.52699, −0.620293

It is to be noted that the real zeros of the polynomials mentioned in Table 2 are giving the numerical results for the approximate solutions of the q-Bessel polynomials p s,1/2 (u) = 0, the q-Bessel-Bernoulli polynomials p Bs,1/2 (u) = 0 and q-Bessel-Euler polynomials p E s,1/2 (u) = 0 for s = 1, 2, 3, 4. Further, the distribution of zeros of the polynomials p s,1/2 (u), p Bs,1/2 (u) and p E s,1/2 (u) are shown in Figure 2 for s = 1, 2, 3, 4. Remark 4.1 For the q-Bessel-Appell polynomials p A s,q (u) of degree s, the following important relation is obtained: Number o f (real zeros of p A s,q (u)) = s − Number o f (complex zeros of p A s,q (u)).

(20)

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Subuhi Khan et al. / Procedia Computer Science 152 (2019) 166–171 Author name / Procedia Computer Science 00 (2019) 000–000

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Table 3. Complex zeros of p s,1/2 (u), p Bs,1/2 (u) and p E s,1/2 (u) Degree s

p s,1/2 (u)

p B s,1/2 (u)

p E s,1/2 (u)

1 2 3

Real zero Real zeros −0.6562 − 0.6242i, −0.6562 + 0.6242i −0.3422 + 0.9043i, −0.3422 − 0.9043i

Real zero 0.125 − 0.282158i, 0.125 + 0.282158i 0.145118 − 0.36446i, 0.145118 + 0.36446i −0.802299 − 0.912951i, 0.606987 − 0.50932i, −0.802299 + 0.912951i, 0.606987 + 0.50932i

Real zero Real zeros 0.116312 − 0.237531i, 0.116312 + 0.237531i 0.72208 − 0.698155i, 0.72208 + 0.698155i

4

Fig. 2. Zeros of p s,1/2 (u);

Zeros of p Bs,1/2 (u);

Zeros of p E s,1/2 (u)

The graphs drawn here provides researchers an unrestricted capacity to fabricate visional mathematical examination of the behaviour of p A s,q (u). In [5], it has been shown that the detection of the positive real roots using the bifurcation is possible by introducing the delay term. The discussion related to the zeros of the polynomials considered here may be useful for analyzing biological models expressing in terms of differential equations. The results obtained in this paper have many applications in numerous areas of biomathematics and engineering. References [1] Al-Salam W.A. (1967) “q-Appell polynomials.” Annali di Matematica Pura ed Applicata. 4 (17): 31-45. [2] Andrews G.E., Askey R., and Roy R. (1999) “Special functions”, of Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge. ´ [3] Appell P. (1880) “Sur une classe de polynˆomes.” Annales scientifiques de l’Ecole Normale Sup´erieure. 9: 119-144. [4] Ernst T. (2006) “q-Bernoulli and q-Euler polynomials, an umbral approach.” International Journal of Differential Equation. 1 (1): 31-80. [5] Forde J., Nelson P. (2004) “Applications of Sturm sequences to bifurcation analysis of delay differential equation models.” Journal of Mathematical Analysis and Application. 300: 273-284. [6] Keleshteri M.E., Mahmudov N.I. (2015) “A study on q-Appell polynomials from determinantal point of view.” Applied Mathematics and Computation. 260: 351-369. [7] Riyasat M., Khan S. (2018) “A determinant approach to q-Bessel polynomials and applications.” Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales. Serie A. Matem´aticas. 1-13.