Mellin transforms of generalized fractional integrals and derivatives

Applied Mathematics and Computation xxx (2015) xxx–xxx

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Mellin transforms of generalized fractional integrals and derivatives Udita N. Katugampola Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

a r t i c l e

i n f o

Keywords: Generalized fractional derivative Riemann–Liouville derivative Hadamard derivative Erdélyi–Kober operators Mellin transform Stirling numbers of the 2nd kind Recurrence relations Hidden Pascal triangles

a b s t r a c t We obtain the Mellin transforms of the generalized fractional integrals and derivatives that generalize the Riemann–Liouville and the Hadamard fractional integrals and derivatives. We also obtain interesting results, which combine generalized dr;m operators with generalized Stirling numbers and Lah numbers. For example, we show that d1;1 corresponds to the Stirling numbers of the 2nd kind and d2;1 corresponds to the unsigned Lah numbers. Further, we show that the two operators dr;m and dm;r ; r; m 2 N, generate the same sequence given by the recurrence relation.

Sðn; kÞ ¼

  r X r ðm þ ðm  rÞðn  2Þ þ k  i  1Þri Sðn  1; k  iÞ; i i¼0

0 < k 6 n;

with Sð0; 0Þ ¼ 1 and Sðn; 0Þ ¼ Sðn; kÞ ¼ 0 for n > 0 and 1 þ minfr; mgðn  1Þ < k or k 6 0.   Finally, we define a new class of sequences for r 2 13 ; 14 ; 15 ; 16 ; . . . and in turn show that d1;1 corresponds to the generalized Laguerre polynomials. 2 Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction The Fractional Calculus (FC) is the generalization of the classical calculus to arbitrary orders. The history of FC goes back to seventeenth century, when in 1695 the derivative of order a ¼ 12 was described by Leibniz. Since then, the new theory turned out to be very attractive to mathematicians as well as biologists, economists, engineers and physicists [27]. In [46], Samko et al. provide a comprehensive treatment of the subject. Several different derivatives were studied: Riemann–Liouville, Caputo, Hadamard, Erdélyi–Kober, Grunwald–Letnikov and Riesz are just a few to name [24,46]. In [19], the author provides an extended reference to some of these and other fractional derivatives. In fractional calculus, the fractional derivatives are defined via fractional integrals [24,28,35,46]. According to the literature, the Riemann–Liouville fractional derivative, hence the Riemann–Liouville fractional integral has played a major roles in FC [46]. The Caputo fractional derivative has also been defined via the Riemann–Liouville fractional integral [46]. Another important fractional operator is the Hadamard operator [14,36]. In [7], Butzer, et al. obtained the Mellin transforms of the Hadamard integral and differential operators. The interested reader is referred to, for example, [5–7,21–24,36,46] for further properties of those operators. In [18], the author introduces a new fractional integral, which generalizes the Riemann–Liouville and the Hadamard integrals into a single form. The interested reader is referred, for example, to [4,12,15,17,20,29–34,37,40] for further results

E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2014.12.067 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

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on these and similar operators. In [19], the author introduces a new fractional derivative which generalizes the Riemann– Liouville and the Hadamard fractional derivatives to a single form. The present work is primarily devoted to the Mellin transforms of the generalized fractional operators developed in [18,19]. The paper is organized as follows. In the next section, we give definitions and some basic properties of the fractional integrals and derivatives of various types. In Section 3, we develop the Mellin transforms of the generalized fractional integrals and derivatives. Further, we investigate the relationship between the Mellin transform operator and the generalized Stirling numbers of the 2nd kind. Finally, we introduce a new class of sequences. 2. Definitions The Riemann–Liouville fractional integrals Iaaþ f and Iab f of order a 2 C; ðReðaÞ > 0Þ are defined by [46],

a  Iaþ f ðxÞ ¼

1 CðaÞ

a  Ib f ðxÞ ¼

1 CðaÞ

Z

x

ðx  sÞa1 f ðsÞds;

x > a:

ð2:1Þ

ðs  xÞa1 f ðsÞds;

x < b;

ð2:2Þ

a

and

Z

b

x

respectively, where CðÞ is the Gamma function. These integrals are called the left-sided and right-sided fractional integrals, respectively. When a ¼ n 2 N, the integrals (2.1) and (2.2) coincide with the n-fold integrals [24, chap. 2]. The Riemann–Liouville fractional derivatives (RLFD) Daaþ f and Dab f of order a 2 C; ðReðaÞ P 0Þ are defined by [46],

 a  Daþ f ðxÞ ¼



n  na  d Iaþ f ðxÞ x > a; dx

ð2:3Þ

and

 n  a   na  d Db f ðxÞ ¼  Ib f ðxÞ x < b; dx

ð2:4Þ

respectively, where n ¼ ½ReðaÞ þ 1, which is sometimes denoted also by the ceiling function, dReðaÞe, when there is no room for confusion. Here ½ represents the integer part. For simplicity, from this point onwards, except in few occasions, we consider only the left-sided integrals and derivatives. The interested reader may find more detailed information about the rightsided integrals and derivatives in the references, for example in [24,46]. The next type that we elaborate in this paper is the Hadamard Fractional integral [14,24,46] given by,

Iaaþ f ðxÞ ¼

1 CðaÞ

Z

x



log

a

x a1

s

f ðsÞ

ds

s

;

ReðaÞ > 0; x > a P 0:

ð2:5Þ

The corresponding Hadamard fractional derivative of order a 2 C; ReðaÞ > 0 is given by,

Daaþ f ðxÞ ¼

 n Z x  1 d x naþ1 ds x log f ð sÞ ; s Cðn  aÞ dx s a

x > a P 0:

ð2:6Þ

where n ¼ dReðaÞe. In 1940, important generalizations of the Riemann–Liouville fractional operators were introduced by Erdélyi and Kober [9]. Erdélyi–Kober-type fractional integral and differential operators [9,10,24,25,46,49] are defined by,

Z  q xqðaþgÞ x sqgþq1 f ðsÞ Iaaþ; q; g f ðxÞ ¼ ds 1a CðaÞ a ðxq  sq Þ

ð2:7Þ

for x > a P 0; ReðaÞ > 0 and

  Daaþ; q; g f ðxÞ ¼ xqg

n  1 d a xqðnþgÞ In aþ; q; gþa f ðxÞ q 1 q x dx

ð2:8Þ

for x > a; ReðaÞ P 0; q > 0, respectively. When q ¼ 2; a ¼ 0, the operators are called Erdélyi–Kober operators. When q ¼ 1; a ¼ 0, they are called Kober-Erdélyi or Kober operators [24, p.105]. Extensive treatment of these operators can be found especially in [49]. Further generalizations can be found, for example in [24,46]. In [18], the author introduces an another generalization to the Riemann–Liouville and the Hadamard fractional integral and also provided existence results and semi-group properties. In [19], the author showed that both the Riemann–Liouville and the Hadamard fractional derivatives can be represented by a single fractional derivative operator. These new operators have been defined on the following extended Lebesgue space.

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3. Generalized fractional integration and differentiation As in [21], consider the space X pc ða; bÞ ðc 2 R; 1 6 p 6 1Þ of those complex-valued Lebesgue measurable functions f on ½a; b for which kf kX pc < 1, where the norm is defined by

kf kXpc ¼

Z

b

dt jt f ðtÞj t c

a

!1=p

p

< 1;

ð1 6 p < 1; c 2 RÞ

ð3:1Þ

and for the case p ¼ 1,

kf kX1c ¼ ess supa6t6b ½t c jf ðtÞj;

ðc 2 RÞ:

ð3:2Þ

In particular, when c ¼ 1=p ð1 6 p 6 1Þ, the space X pc coincides with the classical Lp ða; bÞ-space with

kf kp ¼

Z

b

!1=p jf ðtÞjp dt

< 1 ð1 6 p < 1Þ;

ð3:3Þ

a

kf k1 ¼ ess supa6t6b jf ðtÞj ðc 2 RÞ:

ð3:4Þ

Here we give the generalized forms of fractional integrals introduced in [18] with a slight modification in the notations. Definition 3.1 (Generalized Fractional Integrals). Let X ¼ ½a; b ð1 < a < b < 1Þ be a finite interval on the real axis R. The generalized left-sided fractional integral q Iaaþ f of order a 2 C ðReðaÞ > 0Þ of f 2 X pc ða; bÞ is defined by

q

 q1a I aaþ f ðxÞ ¼ CðaÞ

Z

x

a

sq1 f ðsÞ ds ðxq  sq Þ1a

ð3:5Þ

for x > a and ReðaÞ > 0. The right-sided generalized fractional integral q Iab f is defined by

q

 q1a I ab f ðxÞ ¼ CðaÞ

Z x

b

sq1 f ðsÞ ds ðsq  xq Þ1a

ð3:6Þ

for x < b and ReðaÞ > 0. When b ¼ 1, the generalized fractional integral is called a Liouville-type integral. Remark 1. It can be seen that these operators are not equivalent to Erdélyi–Kober operators due to the presence of a factor

qa , which is essential in the case of Hadamard operators. Now consider the generalized fractional derivatives introduced in [19]; Definition 3.2 (Generalized fractional derivatives). Let a 2 C; ReðaÞ P 0; n ¼ ½ReðaÞ þ 1 and q > 0. The generalized fractional derivatives, corresponding to the generalized fractional integrals (3.5) and (3.6), are defined, for 0 6 a < x < b 6 1, by

q

 n  n Z x  q na  d qanþ1 d sq1 f ðsÞ Daaþ f ðxÞ ¼ x1q I aþ f ðxÞ ¼ x1q anþ1 ds dx dx Cðn  aÞ a ðxq  sq Þ

ð3:7Þ

q

 n  n Z b  q na  d qanþ1 d sq1 f ðsÞ Dab f ðxÞ ¼ x1q I b f ðxÞ ¼ x1q anþ1 ds; q dx dx Cðn  aÞ x ð s  xq Þ

ð3:8Þ

and

respectively, if the integrals exist. When b ¼ 1, the generalized fractional derivative is called a Weyl or Liouville-type derivative [19]. Next we give several important theorems related to these operators, without proofs. Interested reader may find them in [19]. The following theorem gives the relations of the generalized fractional derivatives to that of Riemann–Liouville and Hadamard. For simplicity, we give only the left-sided versions here. The similar results exist for the right-sided operators as well. Theorem 3.3 [19]. Let a 2 C; ReðaÞ P 0; n ¼ dReðaÞe and q > 0. Then, for x > a,

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Z x 1 ðx  sÞa1 f ðsÞds; q!1 CðaÞ a Z x   1 x a1 ds log f ðsÞ ; 2: limþ q I aaþ f ðxÞ ¼ s CðaÞ a s q!0  n Z x q a  d 1 f ðsÞ 3: lim Daþ f ðxÞ ¼ ds; q!1 dx Cðn  aÞ a ðx  sÞanþ1  n Z x    1 d x naþ1 ds x log f ð sÞ : 4: limþ q Daaþ f ðxÞ ¼ s Cðn  aÞ dx s q!0 a

  1: lim q I aaþ f ðxÞ ¼

ð3:9Þ ð3:10Þ ð3:11Þ ð3:12Þ

Proof. We reproduce the proof here for completeness. The Eqs. (3.9) and (3.11) follow from taking limits when q ! 1, while (3.10) follows from the L’Hôspital rule by noticing that [18]

limþ

q!0

q1a CðaÞ

Z

x a

f ðsÞsq1 ðxq  sq Þ1a

 q a1 Z x 1 x  sq ds f ðsÞ limþ s q CðaÞ a q q!0 a q!0  a1 Z x Z x q q a 1 1 x s ds 1 x ds ¼ f ðsÞ limþ ¼ log f ðsÞ ; s CðaÞ a s CðaÞ a q s q!0

ds ¼

1 CðaÞ

Z

x

limþ f ðsÞsq1

 q a1 x  sq

ds ¼

ð3:13Þ

taking into account the principal branch of the logarithm in Eq. (3.13). The proof of (3.12) is similar. Remark 2. Note that the Eqs. (3.9) and (3.11) are related to the Riemann–Liouville operators, while the Eqs. (3.10) and (3.12) are related to the Hadamard operators. The results similar to (3.9) and (3.11) can also be derived from Erdélyi–Kober operators, though it is not possible to obtain equivalent results for (3.10) and (3.12) due to the absence of the factor qa , which is necessary in the case of the Hadamard-type operators. Next is the inverse property. Theorem 3.4 [19]. Let 0 < a < 1, and f 2 X pc ða; bÞ. Then, for a > 0; q > 0,

q

 Daaþ q I aaþ f ðxÞ ¼ f ðxÞ:

ð3:14Þ

Compositions between the operators of generalized fractional differentiation and generalized fractional integration are given by the following theorem. Theorem 3.5 [19]. Let a; b 2 C be such that 0 < ReðaÞ < ReðbÞ < 1. If 0 < a < b < 1 and 1 6 p 6 1, then, for f 2 Lp ða; bÞ; q > 0, q

a Daaþ q I baþ f ¼ q I b aþ f

and

q

a Dab q I bb f ¼ q I b b f :

In [19], the author derived the formula for the generalized derivative (3.7) of the power function f ðxÞ ¼ xm , where for a 2 Rþ ; 0 < a < 1 and a ¼ 0, given by,



q a

D0þ xm ¼



C 1 þ qm qa1 

m 2 R,

C 1 þ qm  a

xmaq

ð3:15Þ

for q > 0. When q ¼ 1, this agrees with the standard results obtained for the Riemann–Liouville fractional derivative (2.3) [24,36,46]. Interestingly enough, for a ¼ 1; q ¼ 1, we obtain 1 D10þ xm ¼ m xm1 , as one would expect. Fig. 1 shows the comparison results of the generalized derivative (3.15) for different values of q ðfor a ¼ 1=2Þ; m ðfor a ¼ 1=2Þ and a. In the next section we give the main results of this paper, the Mellin transforms of the generalized fractional operators. 4. Mellin transforms of generalized fractional operators According to Flajolet et al. [11], Hjalmar Mellin1 (1854–1933) gave his name to the Mellin transform that associates to a function f ðxÞ defined over the positive reals, the complex function M½f  [26]. It is closely related to the Laplace and Fourier transforms. We start by recalling the important properties of the Mellin transform. The domain of definition is an open strip, ha; bi, say, of complex numbers s ¼ r þ it such that 0 6 a < r < b. Here we adopt the definitions and properties mentioned in [11] with some minor modifications to the notations. 1

H. Mellin was a Finnish and his adviser was Gosta Mittag–Leffler, a Swedish mathematician. Later he also worked with Kurt Weierstrass [48].

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Fig. 1. Generalized fractional derivative of the power function xm for q ¼ 0:4; 1:0; 1:4; m ¼ 0:5; 2:0 and a ¼ 0:1; 0:5; 0:9 [19]

Definition 4.1 (Mellin transform). Let f ðxÞ be locally Lebesgue integrable over ð0; 1Þ. The Mellin transform of f ðxÞ is defined by

M½f ðsÞ ¼

Z

1

xs1 f ðxÞ dx:

0

The largest open strip ha; bi in which the integral converges is called the fundamental strip. Following theorem will be of great importance in applications. Theorem 4.2. [11, Theorem 1] Let f ðxÞ be a function whose transform admits the fundamental strip ha; bi. Let q be a nonzero real number, and l; m be positive reals. Then,

" # X 1: M kk f ðlk xÞ ðsÞ ¼ k

X kk k2I

lsk

! M½f ðsÞ;

I finite; kk > 0; s 2 ha; bi

2: M½xm f ðxÞðsÞ ¼ M½f ðs þ mÞ s 2 ha; bi   1 s ; s 2 hqa; qbi 3: M½f ðxq ÞðsÞ ¼ M½f  q q

 d f ðxÞ ðsÞ ¼ ð1  sÞ M½f ðs  1Þ 4: M dx

 d 5: M x f ðxÞ ðsÞ ¼ s M½f ðsÞ dx

Z x  1 6: M f ðtÞdt ðsÞ ¼  M½f ðs þ 1Þ s 0 Next is the inversion theorem for Mellin transforms.

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Theorem 4.3. [11, Theorem 2] Let f ðxÞ be integrable with fundamental strip ha; bi. If c is such that a < c < b, and M½f ðc þ itÞ integrable, then the equality,

1 2pi

Z

c þ i1

M½f ðsÞ xs ds ¼ f ðxÞ c  i1

holds almost everywhere. Moreover, if f ðxÞ is continuous, then equality holds everywhere on ð0; 1Þ. Following is the first important result of the paper, i.e., the Mellin transforms of the generalized fractional integrals. We give both the left-sided and right-sided versions of the results here. Lemma 4.4. Let a 2 C; ReðaÞ > 0, and q > 0. Then,

 C 1  qs  a q a  M I aþ f ðsÞ ¼  Mf ðs þ aqÞ; Reðs=q þ aÞ < 1; x > a; C 1  qs qa  C qs q a  Mf ðs þ aqÞ; Reðs=qÞ > 0; x < b; M I b f ðsÞ ¼  C qs þ a qa

ð4:1Þ

ð4:2Þ

for f 2 X 1sþaq ðRþ Þ, if Mf ðs þ aqÞ exists for s 2 C. Proof. We use the Fubini’s theorem and the Dirichlet technique here. By the Definition 4.1 and Eq. (3.5), we have

  M q I aaþ f ðsÞ ¼ ¼

Z

1

xs1

0 1a

q

CðaÞ

Z

q1a CðaÞ

1

Z

x

a1 q1

ðxq  sq Þ

a

sq1 f ðsÞ

Z

1

s

0

Z

s

f ðsÞ ds dx;

xs1 ðxq  sq Þ

a1

dx ds

Z

1 s qa 1 sþaq1 s f ðsÞ uqa ð1  uÞa1 du ds CðaÞ 0 0  qa C 1  qs  a Z 1  ¼ ssþaq1 f ðsÞ

¼

0 C 1  qs  C 1  qs  a M½f ðs þ aqÞ for Reðs=q þ aÞ < 1: ¼  C 1  qs qa

after using the change of variable u ¼ ðs=xÞq , and properties of the Beta function. This proves (4.1). The proof of (4.2) is similar. Remark 3. The two transforms above confirm Lemma 2.15 of [24] for q ¼ 1. For the case, when q ! 0þ , consider the quotient expansion of two Gamma functions at infinity given by [10],



  ;

Cðz þ aÞ 1 ¼ zab 1 þ O z Cðz þ bÞ Then, it is clear that,

lim

q!0þ

q

a

jargðz þ aÞj < p; jzj ! 1:





I aþ f ðxÞ ¼ limþ

C 1  qs  a

q!0





ð4:3Þ



C 1  qs qa

M½f ðs þ aqÞ ¼ ðsÞa M½f ðsÞ;

ReðsÞ < 0;

where ðsÞa ¼ ea logðsÞ , taking into account the principal branch of the logarithm. This confirms Lemma 2.38(a) of [24]. To prove the next result we use the Mellin transform of mth derivative of a m-times differentiable function given by,

Lemma 4.5. [24, p. 21] Let u 2 Cm ðRþ Þ; M½uðtÞðs  mÞ and M½Dm uðtÞðsÞ exist, and limt!0þ tsk1 uðmk1Þ ðtÞ and sk1 ðmk1Þ limt!þ1 t u ðtÞ are finite for k ¼ 0; 1; . . . ; m  1; m 2 N, then

M½Dm uðtÞðsÞ ¼

m1 X Cð1 þ k  sÞ

1 Cð1 þ m  sÞ xsk1 uðmk1Þ ðxÞ 0 M½uðs  mÞ þ Cð1  sÞ C ð1  sÞ k¼0

M½Dm uðtÞðsÞ ¼ ð1Þm

CðsÞ Cðs  mÞ

M½uðs  mÞ þ

m1 X

ð1Þk

k¼0

CðsÞ sk1 ðmk1Þ 1 x u ðxÞ 0 Cðs  kÞ

ð4:4Þ

ð4:5Þ

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The next result is the Mellin transforms of the generalized fractional derivatives. For simplicity we consider only the case 0 < a < 1 here. In this case, n ¼ dae ¼ 1. Theorem 4.6. Let a 2 C; ReðaÞ > 0; s 2 C; q > 0 and f ðxÞ 2 X 1saq ðRþ Þ. Also assume f ðxÞ satisfies the following conditions:

   a  sq q 1a lim xsq q I 1 I aþ f ðxÞ ¼ 0; aþ f ðxÞ ¼ lim x

x!0þ

and

x!1

   a  sq q 1a lim xsq q I 1 I b f ðxÞ ¼ 0; b f ðxÞ ¼ lim x

x!0þ

x!1

respectively. Then,

 qa C 1  qs þ a q a   M Daþ f ðsÞ ¼ M½f ðs  aqÞ; Reðs=qÞ < 1; x > a P 0; C 1  qs  qa C qs q a  M½f ðs  aqÞ; Reðs=q  aÞ > 0; x < b 6 1; M Db f ðsÞ ¼  C qs  a

ð4:6Þ

ð4:7Þ

respectively. Proof. By Definition 4.1 and Eq. (3.7), we have

 Z x d sq1 xs1 x1q f ðsÞ ds dx; q dx a ðx  sq Þa Cð1  aÞ 0 Z 1 d q 1a  xsq I aþ f ðxÞdx ¼ dx 0  qa C 1  qs þ a   1 a  M½f ðs  aqÞ þ xsq q I 1 ¼ aþ f ðxÞja s C 1q

  M q Daaþ f ðsÞ ¼

qa

Z

1

for Reðs=qÞ < 1, using (4.4) with m ¼ 1 and (3.5). This establishes (4.6). The proof of (4.7) is similar.

ð4:8Þ

h

Remark 4. The Mellin transforms of Riemann–Liouville and Hadamard fractional derivatives follow easily, i.e., the transforms above confirm Lemma 2.16 of [24] for q ¼ 1; b ¼ 1 and Lemma 2.39 of [24] for the case when q ! 0þ in view of (4.3). Remark 5. In many cases we only considered left-sided derivatives and integrals. But the right-sided operators can be treated similarly. dReðaÞe

d d Remark 6. For ReðaÞ > 1, we need to replace x1q dx in (4.8) by ðx1q dx Þ . In this case the equation becomes very complicated and introduces several interesting combinatorial problems. There is a well-developed branch of mathematics called Umbral Calculus [44,45], which consider the case when 1  q 2 N. Bucchianico and Lobe [3] provide an extensive survey of the subject. We discuss some of them here referring interested reader to the works of Rota, Robinson and Roman [39,41–43,45]. First consider the generalized d-derivative defined by,

Definition 4.7. Let r 2 R and m 2 N0 ¼ f0; 1; 2; 3; . . .g. The generalized dr;m -differential operator is defined by m

dr;m :¼ xr th

d m: dx

ð4:9Þ

The n degree dr;m – derivative operator has interesting properties. When r ¼ m ¼ 1 it generates the Stirling numbers of the second kind, Sðn; kÞ [8], Sloane’s A008277 [47], while r ¼ 2 and m ¼ 1 generate the unsigned Lah numbers, Lðn; kÞ, Sloane’s A008297, A105278 [47]. They have been well-studied and appear frequently in literature of combinatorics [2]. These can also be looked at as the number of ways to place j non-attacking rooks on a Ferrer’s-board with certain properties based on r [16]. Now, let cnr;m; j ; j ¼ 1; 2; 3; . . . ; n; n 2 N; r 2 R be the jth coefficient of the expansion of dnr;m , in a basis, n o dm nðk1Þþ2 dmþ1 dnþm B ¼ xnðk1Þþ1 dx ; . . . ; xnk dx m ; x nþm , i.e., dxmþ1

 m n m mþ1 nþm d d d d xk m ¼ cnr;m; 1 xnðk1Þþ1 m þ cnr;m; 2 xnðk1Þþ2 mþ1 þ    þ cnr;m; n xnk nþm dx dx dx dx

ð4:10Þ

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Table 1 Coefficients of nth degree generalized d-derivative.

Table 1 lists the cases for r ¼ 1 and r ¼ 2 when m ¼ 1 in triangular settings, The Stirling numbers of the second kind, Sðn; kÞ and unsigned Lah numbers, Lðn; kÞ are given by [2]

Sðn; kÞ ¼

k 1X ð1Þi k! i¼0

  k n ðk  iÞ ; i

0 6 k 6 n 2 N [ f0g;

ð4:11Þ

and

Lðn; kÞ ¼



n1 k1



n! ; k!

0 6 k 6 n 2 N [ f0g;

ð4:12Þ

respectively. A closed-form formula exists for the case r ¼ 2, and asymptotic formula exists for r ¼ 1 [13,38]. Johnson [16] used analytical methods and Goldman [13] used algebraic methods to obtain generating functions for r ¼ 1; 2 given by

8  xy  < exp e x1 n o zðx; yÞ ¼ : exp 1 ½1  ðr  1Þxy1=ð1rÞ  1 x x

if r ¼ 1; ð4:13Þ

if r P 2:

m

d It is interesting to note that the order of xr and dx m in the dr;m operator is of no importance as indicated in the following examples, as long as r and m are kept fixed. For example, we consider d2;1 and d1;2 . The first five rows of the corresponding Pseudo polynomials are given in Tables 2 and 3. Ò These polynomials can be generated using the Maple codes given in the Algorithm 1 for r ¼ 2 and m ¼ 1. This interesting result holds for any r 2 N ¼ f1; 2; 3; . . .g and will be given in the following result. In Theorem 4.9, we further show that the dm m dr differential operators xr dx produce the same sequence of coefficients as well (see Table 4). m and x dxr

Theorem 4.8. Let r 2 N ¼ f1; 2; 3; . . .g. The generalized Stirling numbers generated by the operators dr;1 and d1;r are coincide with each other and follow the recurrence relation given by

Sðn; kÞ ¼ ½ðn  1Þðr  1Þ þ kSðn  1; kÞ þ Sðn  1; k  1Þ;

0 6 k 6 n;

ð4:14Þ

with Sð0; 0Þ ¼ 1 and Sðn; 0Þ ¼ Sðn; kÞ ¼ 0 for k P n þ r P 0. It is clear from (4.14) that when r ¼ 1, the recurrence relation coincide with that of the Stirling numbers of the second kind and when r ¼ 2, it coincide with that of the unsigned Lah numbers. Proof. We prove Theorem 4.8 by induction on n and k for dr;1 . For n ¼ k ¼ 1, we have Sð1; 1Þ ¼ Sð0; 1Þ þ Sð0; 0Þ ¼ 1. Now, for an inductive argument, assume the recurrence relation (4.14) holds for n and k and prove the relation holds for n þ 1 and d d k þ 1. First consider the case Sðn þ 1; 1Þ. The coefficient of xrþðr1Þðn1Þ dx is Sðn; 1Þ and applying the operator xr dx , we see that Sðn þ 1; 1Þ ¼ ðr þ ðr  1Þðn  1ÞÞSðn; 1Þ. For Sðn þ 1; n þ 1Þ, notice that Sðn; n þ 1Þ ¼ 0 and Sðn; nÞ ¼ 1 and by applying the d , we have Sðn þ 1; n þ 1Þ ¼ Sðn; nÞ. For k R f1; n þ 1g, the coefficient Sðn þ 1; kÞ of xrþðr1Þðn1Þþk differential operator xr dx th

obtained by applying dr;1 to two terms in the n

step, namely, Sðn; k  1Þxrþðr1Þðn2Þþk1

k1

d k1 dx

and Sðn; kÞxrþðr1Þðn2Þþk

k

d k dx

dk k dx

is

and

collecting proper terms. In the first product, we differentiate the second term, thus keep the coefficient fixed. From the second product, we obtain Sðn; kÞðr þ ðr  1Þðn  2Þ þ kÞ. This proves the Eq. (4.14) for n. Now, for the case k, we use a similar argument. It is clear that Sðk; k þ 1Þ ¼ 0 and Sðk þ 1; k þ 1Þ ¼ 1 for k 2 N. The coefficients Sðk þ 2; k þ 1Þ; Sðk þ 3; k þ 1Þ; . . . ; Sðn; k þ 1Þ can be obtained following the case of n. The case of d1;r is similar. This completes the proof. The Theorem 4.8 can be extended to the cases of any r; m 2 N. Let us first consider, for example, the cases of d2;3 and d3;2 . In both the cases, it generates the same triangular array of sequences given in Table 4. This interesting result holds for any two positive integers and is given in Theorem 4.9.

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U.N. Katugampola / Applied Mathematics and Computation xxx (2015) xxx–xxx Table 2 Pseudo polynomials of d2;1 .

Table 3 Pseudo polynomials of d1;2 .

Table 4 The sequence for d2;3 and d3;2 . n

1 2 3 4 .. .

k cn;1

cn;2

cn;3

cn;4

cn;5

cn;6

cn;7

1 6 72 1440

6 168 5760

1 96 6120 .. .

18 2520

1 456

36

1

cn;8

..

.

4.1. Hidden Pascal triangles Even though the terms in the sequence d3;2 have no apparent resemblance to any familiar sequences, it is interesting to notice that there is a hidden Pascal Triangle buried inside those terms. The reason is that, in each differentiation step, it d creates two terms, one with a degree one less than the degree of x and one with a order one greater than that of dx . Thus, it contributes a term to the left and right of the Pascal triangle in each differentiation step. For example, to obtain the second row of the operator d4;3 , we pass though the steps in Table 5, in that order, after each differentiation step. Table 6 corresponds to the finite sub-Pascal triangle of order four, ðr þ 1Þ, with rows given in Table 5. In the case of d3;4 , there will be no ‘two-pair’ contributions after the third step. Thus, minfr; mg decides the length of the sub-Pascal triangle. This observation extends Theorem 4.8 to the more general version given in the next result. Theorem 4.9. Let r; m 2 N ¼ f1; 2; 3; . . .g and r 6 m. The generalized Stirling numbers generated by the operators dr;m and dm;r are coincide with each other and follow the recurrence relation given by

Sðn; kÞ ¼

  r X r Sðn  1; k  iÞ; ðm þ ðm  rÞðn  2Þ þ k  i  1Þri i i¼0

0 < k 6 n;

ð4:15Þ

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U.N. Katugampola / Applied Mathematics and Computation xxx (2015) xxx–xxx

Table 5 The second row of the d4;3 .

Table 6 sub-Pascal triangle for r ¼ 3.

with Sð0; 0Þ ¼ 1 and Sðn; 0Þ ¼ Sðn; kÞ ¼ 0 for n > 0 and 1 þ minfr; mgðn  1Þ < kor k 6 0. Here ðÞk is the Pochhammer symbol for falling factorials. Sketch of proof. The proof is similar to that of Theorem 4.8 except that, in this case, it is required to consider r þ 1 many terms, which contribute to the coefficient Sðn; kÞ. The appearance of the binomial coefficient is clear from the discussion of Section 4.1. The term ðm þ ðm  rÞðn  2Þ þ k  i  1Þri is obtained after r  i many differentiations of the power of the basic term xmþðmrÞðn2Þþki1

dmþki . dxmþki

This completes the proof.

It is clear from Eq. (4.15) that when r ¼ m ¼ 1, the recurrence relation coincide with that of Stirling numbers of 2nd kind and when r ¼ 2; m ¼ 1, it coincide with that of the unsigned Lah numbers. The MapleÒ implementation of the recurrence relation (4.15) is given in the Algorithm 2. All the sequences for the cases k 2 f1; 2; 3; . . . ; 20g are listed at A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511–A223522 [47]. 5. New class of sequences We now consider rational values of r 2 R. To begin our work, we first consider r ¼ 12 ; 13 ; 14 ; 15 and 16.  n1 pffiffiffi d n x dx Definition 5.1. The generalized Stirling numbers of order 12 is defined by the coefficients of 2 2 , when n is odd-positive, and  n pffiffiffi n d of 22 x dx , when n is even-positive. The sequence corresponding to the Definition 5.1 is given below, (see Table 7), Sloane’s A223168 [47].

Table 7 The sequence for d1;1 . 2

n

1 2 3 4 5 6 7 8 9 .. .

k cn;1

cn;2

1 1 3 3 15 15 105 105 945

2 2 12 20 90 210 840 2520

cn;3

cn;4

cn;5

4 4 60 84 840 1512 .. .

8 8 224 288

16 16

cn;6

..

.

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U.N. Katugampola / Applied Mathematics and Computation xxx (2015) xxx–xxx Table 8 Pseudo polynomials for the operator d1;1 . 2

pffiffiffi d 1 20 x dx pffiffiffi d 2 21 x dx pffiffiffi d 3 21 x dx 2 pffiffiffi d 4 2 x dx pffiffiffi d 5 22 x dx

¼ 1x

1 2

d , dx

d ¼ 1 dx þ 2x

¼ 3x ¼3

2

1 2 2

d 2 dx

¼ 15x

1 2

2

d 2, dx 3

d dx2

þ 2x2

þ 12x 3

d 3 dx

3

d , dx3 3

d 3 dx

þ 20x

þ 4x2 3 2

4

d 4 dx

d4 4, dx 5

þ 4x2

5

d , dx5

.. .

The expansion of this operator takes the form given in Table 8. These pseudo polynomials may be generated by the Maple code listed in Algorithm 5. According to V. Kotesovec [47, A223168], the coefficients of the kth row of d1;1 is given by the kth 2

derivative of

1 dk=2e

2

2

ex , where de is the ceiling function. The odd and even rows of this triangular sequence are listed at

A223523 and A223524 [47]. Odd rows have further interesting properties. For example, it includes absolute values of A098503 [47] from right to left, which are coefficients of the generalized Laguerre polynomials up to an integer multiple of 2n for some n 2 N. They are given by the Rodrigues formula [1], n

LðnaÞ ðxÞ ¼

xa ex d ðex xnþa Þ; n! dxn

which are solutions of the differential equation,

x y00 þ ða þ 1  xÞ y0 þ n y ¼ 0; for arbitrary a. The first few generalized Laguerre polynomials are given by, ðaÞ

L0 ðxÞ ¼ 1; ðaÞ

L1 ðxÞ ¼ x þ a þ 1; x2 ða þ 2Þða þ 1Þ  ða þ 2Þx þ ; 2 2 3 2 x ða þ 3Þx ða þ 2Þða þ 3Þx ða þ 1Þða þ 2Þða þ 3Þ ðaÞ þ  þ ; L3 ðxÞ ¼ 2 6 6 2 ðaÞ

L2 ðxÞ ¼

and the generalized Laguerre polynomial of degree n is given by [1]

LðnaÞ ðxÞ ¼

n  n þ a xi X ð1Þi : n  i i! i¼0

It can easily be seen that when a ¼ 1=2, we obtain A223523 [47] up to signs of the terms. 5.1. Generalized d sequences The Definition 5.1 can be extended for a large class of rational numbers. Let us now define dr;1 sequences for other values of r as well.    1n  d n Definition 5.2. The generalized Stirling numbers of order r 2 12 ; 13 ; 14 ; . . . , is defined by the coefficients of r 2 xr dx , when n  n n 2 1r d is odd-positive, and the coefficients of r x , when n is even-positive. dx The corresponding sequence for r 2 f13 ; 14 ; 15 ; 16g are listed in Table 9. All of these sequences and odd and even rows of them are listed at Sloane’s A223168–A223172 and A223523–A223532 [47]. Remark 7. The combinatorial interpretation of these sequences are unknown at the time of the production of this paper and the author suggests that they may be related to counting on Ferrer’s-board, especially to r-creation boards. The author’s future work will be focused on the properties of these sequences and the representations of such sequences and beyond.

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U.N. Katugampola / Applied Mathematics and Computation xxx (2015) xxx–xxx

Table 9 The generalized Stirling numbers for r ¼ 13 ; 14 ; 15, and 16. n

k

(a) r ¼ 1 2 3 4 5 6 7 8 9 .. .

1 3

(b) r ¼ 1 2 3 4 5 6 7 8 9 .. .

1 4

cn;1

cn;2

1 1 4 4 28 28 280 280 3640

3 3 24 42 252 630 3360 10920

1 1 5 5 45 45 585 585 9945

(c) r ¼ 15 1 2 3 4 5 6 7 8 9 .. .

1 1 6 6 66 66 1056 1056 22176

(d) r ¼ 16 1 2 3 4 5 6 7 8 9 .. .

1 1 7 7 91 91 1729 1729 43225

4 4 40 72 540 1404 9360 31824

5 5 60 110 990 2640 21120 73920

6 6 84 156 1638 4446 41496 148200

cn;3

cn;4

cn;5

9 9 189 270 3780 7020 .. .

27 27 1080 1404

81 81

16 16 432 624 11232 21216 .. .

64 64 3328 4352

25 25 825 1200 26400 50400 .. .

125 125 8000 10500

36 36 1404 2052 53352 102600 .. .

216 216 16416 21600

cn;6

..

.

..

.

..

.

..

.

256 256

625 625

1296 1296

Remark 8. It would be very interesting if one could classify all or a large class of such dr;m sequences. This could be a direction for future research. We would like to represent Sðn; kÞ as d1;1 -type and Lðn; kÞ as d2;1 -type, and etc. We may even consider negative integer or irrational values of r 2 R. To the author’s knowledge this would lead to a new classification or at least a new method to represent some familiar sequences in combinatorial theory.

6. Conclusion The paper presents the Mellin transforms of the generalized fractional integrals and derivatives, which generalize the Riemann–Liouville and the Hadamard fractional operators. We also show that the dr;l operator corresponds to several familiar sequences, specially generalized Stirling numbers and Lah numbers. We also show that the two operators dr;m and dm;r ; r; m 2 N, generate the same sequence. Further, we obtain the recurrence relations for the generalized Stirling numbers generated by the differential operators dr;m for r; m 2 N and finally,

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U.N. Katugampola / Applied Mathematics and Computation xxx (2015) xxx–xxx

13

we define a new class of sequences for r 2 f12 ; 13 ; 14 ; 15 ; . . .g and in tern show that d1;1 corresponds to the generalized Laguerre 2 polynomials. In a future project, we will derive formulae for the Laplace and Fourier transforms for the generalized fractional operators. We want to further investigate the effect on the new control parameter q, which makes more sense in this new setting. Acknowledgments The author would like to thank Philip Feinsilver, the Department of Mathematics at Southern Illinois University, for pointing out interesting results related to Umbral Calculus, Darin B. Johnson, the Department of Mathematics at Delaware State University, for pointing out interesting connections to some literature in combinatorial theory and Jerzy Kocik, the Department of Mathematics at Southern Illinois University for valuable comments and suggestions. The author also would like to thank the anonymous referees for their invaluable suggestions, which put the article in its present form. Appendix This section contains the required MapleÒ codes that generate the sequences we discuss throughout the paper. Some of these codes are also listed in the Sloane’s On-line Encyclopedia of Integer Sequences at http://oeis.org under the sequences A223168–A223172 and A223523–A223532  [47]. n dm The pseudo polynomials of the operator xr dx for r; m 2 N are generated by the Maple code in Algorithm 1. m Algorithm 1. a[0] :¼ f(x): for i to 10 do a[i] :¼ simplify(x^r⁄diff(a[i-1],x$m)); end do

The generalized Stirling numbers Sðn; kÞ can be generated by the following Maple recursive procedure. Algorithm 2. genStrl :¼ proc (n::integer, k::integer, r1::nonnegint, m1::nonnegint) local r, m, i, total :¼ 0; option remember; r :¼ min(r1, m1); m :¼ max(r1,m1); if n <= 0 or k <= 0 or 1 + r⁄(n-1) < k then return 0; elif n = 1 and k = 1 then return 1 else for i from 0 to r do total :¼ total+ mul(m+(m-r)⁄(n-2)+k-i-1-j, j = 0 .. r-i-1) ⁄ binomial(r, i)⁄genStrl(n-1, k-i, r, m); end do; return total; end if; end proc:

The pseudo polynomials for r ¼ 12 are generated by the Maple code given below. Algorithm 3. a[0] :¼ f(x): for i from 1 to 10 do a[i]:¼ simplify(2^((i+1) mod 2)⁄x^(1/2)⁄(diff(a[i-1],x$1))); end do; The pseudo polynomials for r ¼ q1 ; q 2

1

;1;1;... 3 4 5



are generated by the Maple code in the Algorithm 4.

Algorithm 4. a[0] :¼ f(x): for i from 1 to 10 do a[i]:¼ simplify(r^((i+1) mod 2)⁄x^(((r-2)⁄((i+1)mod 2)+1)/r) ⁄(diff(a[i-1],x$1))); end do;

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U.N. Katugampola / Applied Mathematics and Computation xxx (2015) xxx–xxx

The following recurrence relation based algorithm generates the generalized sequences in a triangular array. Here we give the case for r ¼ 14. Other cases are similar. Algorithm 5. n :¼ 10; r :¼ 1/4; S :¼ Array(1 .. n, 1 .. n); # This generates a nxn array of zeros S[1, 1] :¼ 1; for i from 2 to n do if i mod 2 =1 then for j to (i+1)⁄(1/2) do S[i, j] :¼ S[i-1,j]+j⁄S[i-1,j+1] # i is odd end do else for j to (1/2)⁄i do S[i, j] :¼ i⁄(1+(j-1)/r)/(i-2⁄(j-1))⁄S[i-1,j] # i is even end do; S[i,i/2+1] :¼ 1/r⁄S[i-1,i/2] # For the case j = i/2+1 end if end do

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, 1965. M. Bóna, Introduction to Enumerative Combinatorics, first ed., McGraw-Hill, New York, 2007. A.D. Bucchianico, D. Loeb, A selected survey of umbral calculus, Elec. J. Combin. 3: Dyn. Surv. Sec. (1995) #DS3. A.G. Butkovskii, S.S. Postnov, E.A. Postnova, Fractional integro-differential calculus and its control-theoretical applications I – mathematical fundamentals and the problem of interpretation, Autom. Rem. Control 74 (4) (2013) 543–574. P. Butzer, A. Kilbas, J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002) 387–400. P. Butzer, A. Kilbas, J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002) 1–27. P. Butzer, A. Kilbas, J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl. 270 (2002) 1–15. P. Butzer, A. Kilbas, J. Trujillo, Stirling functions of the second kind in the setting of difference and fractional calculus, Numer. Funct. Anal. Opt. 24 (7&8) (2003) 673–711. A. Erdélyi, H. Kober, Some remarks on Hankel transforms, Q. J. Math. Oxford Ser. 11 (1940) 212–221. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. 1, Krieger Pub., Florida, 1981. I–III. P. Flajolet, X. Gourdon, P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theor. Comput. Sci. 144 (1–2) (1995) 3–58. S. Gaboury, R. Tremblay, B. Fugre, Some relations involving a generalized fractional derivative operator, J. Inequal. Appl. 2013 (2013) 167. J. Goldman, J. Joichi, D. White, Rook Theory I, Rook equivalence on Ferrers boards, Proc. Am. Math. Soc. 52 (1975) 485–492. J. Hadamard, Essai sur l’etude des fonctions donnees par leur developpment de Taylor, J. Pure Appl. Math. 4 (8) (1892) 101–186. R. Herrmann, Fractional Calculus: An Introduction for Physicists, second ed., World Scientific, River Edge, New Jersey, 2014. D.B. Johnson, Topics in Probabilistic Combinatorics (Ph.D. Dissertation), Southern Illinois University, Carbondale, 2009. U.N. Katugampola, On Generalized Fractional Integrals and Derivatives (Ph.D. Dissertation), Southern Illinois University, Carbondale, 2011. U.N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218 (3) (2011) 860–865. U.N. Katugampola, New approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (4) (2014) 1–15. Availabel from: arXiv:1106.0965v5. U.N. Katugampola, Existence and uniqueness results for the generalized fractional differential equations, in: Proceedings of the 5th IFAC Symposium on Fractional Differentiation and Its Applications, Nanjing, China, 2012. A.A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (6) (2001) 1191–1204. A.A. Kilbas, M. Saigo, H-Transforms: Theory and Applications, Chapman & Hall/CRC, New York, 2004. A.A. Kilbas, J.J. Trujillo, Hadamard-type integrals as G-transforms, Integral Trans. Spec. Funct. 14 (5) (2003) 413–427. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B.V, Amsterdam, Netherlands, 2006. V. Kiryakova, Generalized Fractional Calculus and Applications, John Wiley & Sons Inc., New York, 1994. E. Lindelöf, Robert Hjalmar Mellin, Adv. Math. 61(i-vi) (1933), (H. Mellin’s note’s and bibliography). J.A. Machado, And I say to myself: what a fractional world!, Frac. Calc. Appl. Anal. 14 (4) (2011) 635–654. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. M.A. Noor, M.U. Awan, K.I. Noor, On Some Inequalities for Relative Semi-Convex Functions, J. Inequal. Appl. 2013 (2013) 332. M.A. Noor, K.I. Noor, M.U. Awan, Generalized convexity and integral inequalities, Appl. Math. Inf. Sci. 9 (1) (2015) 233–243. M.A. Noor, K.I. Noor, M.U. Awan, S. Khan, Fractional Hermite–Hadamard inequalities for some new classes of Godunova–Levin functions, Appl. Math. Inf. Sci. 8 (6) (2014) 2865–2872. T. Odzijewicz, A. Malinowska, D. Torres, A generalized fractional calculus of variations, Control Cybern. 42 (2) (2013) 443–458. T. Odzijewicz, A. Malinowska, D. Torres, Fractional calculus of variations in terms of a generalized fractional integral with applications to physics, Abstr. Appl. Anal. 2012 (2012). id-871912. T. Odzijewicz, Generalized fractional Isoperimetric problem of several variables, Discrete Contin. Dyn. S. Ser. B 19 (8) (2014) 2617–2629. K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, California, U.S.A, 1999. S. Pooseh, R. Almeida, D. Torres, Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Numer. Funct. Anal. Opt. 33 (3) (2012) 301–319.

Please cite this article in press as: U.N. Katugampola, Mellin transforms of generalized fractional integrals and derivatives, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2014.12.067

U.N. Katugampola / Applied Mathematics and Computation xxx (2015) xxx–xxx [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49]

15

H. Robbins, A remark on Stirling’s formula, Amer. Math. Mon. 62 (1955) 26–29. T.J. Robinson, Formal calculus and umbral calculus, Elec. J. Combin. 17 (2010). #R95. M.M. Rodrigues, Generalized fractional integral transform with Whittaker’s kernel, AIP Conf. Proc. 1561 (2013) 194. S. Roman, The theory of the umbral calculus I, J. Math. Anal. Appl. 87 (1) (1982) 58–115. S. Roman, The theory of the umbral calculus II, J. Math. Anal. Appl. 89 (1) (1982) 290–311. S. Roman, The theory of the umbral calculus III, J. Math. Anal. Appl. 95 (2) (1983) 528–563. G.C. Rota, Finite Operator Calculus, Acamedic Press, New York, 1975. G.C. Rota, S. Roman, The Umbral Calculus, Adv. Math. 27 (1978) 95–188. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, 1993. Yverdon et alibi. N.J.A. Sloane, The On-line Encyclopedia of Integer Sequences, , 2014. The MacTutor History of Mathematics Archive, , 2014. S.B. Yakubovich, Y.F. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions, Kluwer Academic, Boston, 1994.

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