Journal of Computational and Applied Mathematics 259 (2014) 108–116 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam A set of finite order differential equations for the Appell polynomials Mehmet Ali Özarslan ∗ , Banu Yılmaz Eastern Mediterranean University, Faculty of Arts and Sciences, Department of Mathematics, Gazimagusa, Mersin 10, Turkey article info Article history: Received 14 September 2012 Received in revised form 25 March 2013 MSC: 33C45 33C55 abstract Let {Rn (x)}∞ n=0 denote the set of Appell polynomials which includes, among others, Hermite, Bernoulli, Euler and Genocchi polynomials. In this paper, by introducing the generalized   ( x) factorization method, for each k ∈ N, we determine the differential operator Ln,k such that ∞ n=0 (x) Ln,k (Rn (x)) = λn,k Rn (x), (n+k)! where λn,k = n! − k!. The special case k = 1 reduces to the result obtained in [M.X. He, P.E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2002) 231–237]. The differential equations for the Hermite and Bernoulli polynomials are exhibited for the case k = 2. © 2013 Elsevier B.V. All rights reserved. Keywords: Appell polynomials Hermite polynomials Bernoulli polynomials Differential equation 1. Introduction Let {Pn (x)}∞ n=0 denote a sequence of polynomials such that deg(Pn ) = n (n ∈ N0 := {0, 1, 2, . . .}) . The differential operators Θn− and Θn+ satisfying the properties Θn− (Pn (x)) = Pn−1 (x), Θn+ (Pn (x)) = Pn+1 (x) are called derivative and multiplicative operators, respectively. The polynomial sequence {Pn (x)}∞ n=0 is called quasimonomial if and only if the above conditions are satisfied. For a given polynomial sequence, Youssèf Ben Cheikh proved the existence of these operators and gave an affirmative answer to Dattoli’s question ‘‘May all polynomial families be viewed as quasi-monomial’’ [1]. More precisely he has shown that ‘‘every polynomial set is quasi-monomial’’ [2]. By using the monomiality principle, several results were obtained for Laguerre, Laguerre–Konhauser, Legendre, Bernoulli and Appell polynomials (see [3–5,1,6–9]). On the other hand orthogonality of some polynomial sets via quasi-monomiality was given in [10]. Obtaining the derivative and multiplicative operator of a given family of polynomials gives rise to some useful properties such as  −  Θn+1 Θn+ (Pn (x)) = Pn (x),  +  Θn−1 · · · Θ2+ Θ1+ Θ0+ (P0 (x)) = Pn (x). ∗ Corresponding author. Tel.: +90 392 630 1266; fax: +90 392 365 1604. E-mail addresses: mehmetali.ozarslan@emu.edu.tr, mozarslan76@gmail.com (M. Ali Özarslan), banu.yilmaz@emu.edu.tr (B. Yılmaz). 0377-0427/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cam.2013.08.006 (1) M. Ali Özarslan, B. Yılmaz / Journal of Computational and Applied Mathematics 259 (2014) 108–116 109 Note that if Θn− and Θn+ are differential realizations, then (1) gives the differential equation satisfied by Pn (x). The technique in obtaining differential equations via (1) is known as the factorization method. In this paper we mainly focus on the Appell polynomial set {Rn (x)}∞ n=0 . The Appell polynomial set is defined by the generating relation [11] A(t )ext = ∞  Rn (x) n=0 tn n! , where A(t ) = ∞  an n=0 tn n! (A(0) ̸= 0) , is an analytic function at t = 0. The Appell polynomials include many important polynomials, for instance: • Taking A(t ) = 1, we get the monomials Rn (x) = xn . t2 • If A(t ) = e− 2 , then Rn (x) = Hen (x),the Hermite polynomials [12,13]. • Choosing A(t ) = λett−1 (|t | < 2π when λ = 1; |t | < |log λ| when λ ̸= 1), then Rn (x) = Bn (x), the Apostol–Bernoulli polynomials [14–16]. Note that when λ = 1, we have the Bernoulli polynomials [17]. • Letting A(t ) = (1 − t )−α (|t | < 1), then Rn (x) = n!Ln(α−n) (x), the modified Laguerre polynomials [18]. m • By taking A(t ) = eht , then Rn (x) = gnm (x, h) the Gould–Hopper polynomials [19]. 2 (|t | < π when λ = 1; |t | < | log(−λ)| when λ ̸= 1), then Rn (x) = En (x), the Apostol–Euler polynomials (see [20–23]). Note that when λ = 1, we have the Euler polynomials [17]. • Choosing A(t ) = λet +1 • Putting A(t ) = λe2t t +1 (|t | < | log(−λ)|), then Rn (x) = Gn (x), the Apostol–Genocchi polynomials [24–26,23,22]. The case λ = 1 gives the Genocchi polynomials.  m αi t • Letting A(t ) = i=1 eαi t −1 (|αi t | < 2π ), then Rn (x) is the Bernoulli polynomials of order m [4]. Note that, when αi = 1 (i = 1, . . . , m) then these polynomials are called Barnes polynomials.  2 • Taking A(t ) = m i=1 eαi t +1 (|αi t | < π ),then Rn (x) is the Euler polynomials of order m [4]. d+1 ξi t i (ξd+1 ̸= 0), then Rn (x) is the generalized Gould–Hopper polynomials [1]. These polynomials include the Hermite polynomials when d = 1 and d-orthogonal polynomials for each positive integer d. • Choosing A(t ) = e i=0 −(k) In this paper, for each k ∈ N we focus on to construct two operators Θn +(k) and Θn satisfying Θn−(k) [Pn (x)] = Pn−k (x) and Θn+(k) [Pn (x)] = Pn+k (x), where we call them the k-times derivative and k-times multiplicative operators, respectively. Obtaining these operators for +(k) −(k) and Θn a given polynomial set will provide us several advantageous relations for that polynomial set. Such as, when Θn are differential operators then, for each k ∈ N, the relation   −(k) Θn+k Θn+(k) (Pn (x)) = Pn (x) (2) gives us differential equations for this polynomial set. In this case we call such a method which is stated by (2) as the generalized factorization method. This method provides us to obtain a set of differential equations for Pn (x), because for each k ∈ N we have one differential equation for this polynomial. On the other hand, if n = mk + r then, by using few number of operators, the second relation in (1) can be given as  +(k) +(k) + Θn+−1 · · · Θmk Θ(m−1)k · · · Θk We organize the paper as follows. +(k) Θ0  (P0 (x)) = Pn (x). −(k) +(k) In Section 2, for each k ∈ N, determining the operators Θn and Θn , we obtain a set of finite order differential equations for Appell polynomials. In Section 3, we give differential equations for the Hermite and Bernoulli polynomials for the case k = 2. 110 M. Ali Özarslan, B. Yılmaz / Journal of Computational and Applied Mathematics 259 (2014) 108–116 2. A set of finite order differential equations for the Appell polynomials via the generalized factorization method −(k) +(k) and Θn In this section by obtaining a recurrence relation for the Appell polynomials, we determine the operators Θn for each k ∈ N. Then using the generalized factorization method, we give a set of finite order differential equations for the Appell polynomials. We exhibit the special cases of our results for k = 1 (the known results) and k = 2. We start with the following theorem. Theorem 2.1. Let A(m) (t ) A(t ) = ∞  αn(m) n=0 tn n! (3) . Then we have the recurrence relation Rn+k (x) = Rn (x)  k   k m m=0 (m) k−m α0 x +  k   k m=1 m xk−m n−1    n l=0 l αn(m−)l Rl (x). (4) Moreover, the k-times shift operators are given by n  Θn−(k) := n  Φm− = m=n−k+1 1 m=n−k+1 m Dx = (n − k)! k Dx n! and +(k) Θn :=  k   k m=0 m k−m x (m) α0 +  k   k m=1 m xk−m n−1  l= 0 1 (n − l)! αn(m−)l Dnx −l . (5) Proof. Let G(x, t ) := A(t )ext = ∞  tl Rl (x) . l! l=0 (6) Differentiating both sides k-times with respect to x, we get ∂ kG = t k G(x, t ). ∂ xk n Using series expansion from (6), in the above relation and equating the coefficients of tn! , we get R(nk) (x) = n! Rn−k (x). (n − k)! (7) Introducing the familiar derivative operator by Φn− = 1 n Dx we see from (7) that Θn−(k) [Rn (x)] := n  Φm− [Rn (x)] = Rn−k (x). m=n−k+1 Then differentiating both sides of (6) k-times with respect to t, we get  k    xt  ∂ kG k k−m e Dm = t {A(t )} Dt k m ∂t m=0   k  k    ∂ mA k A(m) (t ) k xk−m m = G(x, t ) = ext xk−m . m m ∂t A(t ) m=0 m=0 (8) M. Ali Özarslan, B. Yılmaz / Journal of Computational and Applied Mathematics 259 (2014) 108–116 111 Upon using (3) and (6) in (8), we get ∞  Rn+k (x) n= 0 tn = n!  k   k m m=0 =  k   k m m=0 ∞  xk−m αn(m) n=0 ∞ tn  n! l= 0 ∞  n    n xk−m n tl l! αn(m−)l Rl (x) l n=0 l=0 Rl (x) tn n! . Now comparing coefficients of tn! on both sides, we get Rn+k (x) =  k   k m m=0 or equivalently xk−m Rn+k (x) = Rn (x) αn(0) = δn,0 := we get   k   k Rn+k (x) = Rn (x) 1, 0, l l=0 m m=0 Since n    n αn(m−)l Rl (x) (m) xk−m α0 + xk−m n−1    n l l=0 αn(m−)l Rl (x). n=0 otherwise  k   k m m=0 n  m m=0 (m) xk−m α0 +  k   k m m=1 which is (4). On the other hand, since Rl (x) =  k   k Φm− [Rn (x)] = m=l+1 l! n! xk−m n−1    n l l=0 αn(m−)l Rl (x), Dxn−l [Rn (x)] we get from (4) that Rn+k (x) =   k   k m m=0 x k−m (m) α0 +  k   k m=1 m x k−m n−1  l=0 1 (n − l)! Therefore the k-times multiplicative operator is given by (5). αn(m−)l Dxn−l  Rn (x).  The next theorem gives a set of differential equations for Appell polynomials for each k ∈ N. Theorem 2.2. For each k ∈ N and for all n ∈ N, the Appell polynomials Rn (x) satisfy the following set of differential equations: (x) L n ,k (Rn (x)) = where (x) Ln,k =   (n + k)! − k! Rn (x), n!  k    k k! j=1 + j! j xj Djx + m m m=1  n− 1 k   k  m=1  k   k l=0 1 (n − l)! α0(m) k    k (k − m)! j=m j (j − m)! k    k (k − m)! (m) αn−l j j= m (j − m)! j− m x xj−m Djx Dnx −l+j  . Proof. Using the k-times shift operators from Theorem 2.1 and applying the generalized factorization method (2) to Rn (x), we get n! (n + k)! = Dkx n!   k   k m=0 (n + k)!  m (m) k−m α0 x  k   k m=0 m +  k   k m=1 α0(m) k    k j= m j m x k−m n− 1  l= 0 1 (n − l)! Dkx−j xk−m Djx (Rn (x))   αn(m−)l Dnx −l  Rn (x) 112 M. Ali Özarslan, B. Yılmaz / Journal of Computational and Applied Mathematics 259 (2014) 108–116 +  n−1 k   k  m=1 = m  n! 1 (n − l)! l=0 k!Rn (x) + (n + k)!  n−1 k   k  + m=1 m αn(m−)l k    k j 1 (n − l)! j j=m j=1 l=0 k    k αn(m−)l  k −m x  Dnx −l+j   k    k j (Rn (x))  k   k Dkx−j xk Djx (Rn (x)) + j=m = Rn (x). Dkx−j m=1 m α0(m) k    k j=m Dkx−j xk−m Dnx −l+j (Rn (x))     j Dkx−j xk−m Djx (Rn (x))   Therefore n! (n + k)! + k    k k! (k!Rn (x)) + j j=1  n−1 k   k  m m=1 j! 1 (n − l)! l=0  k   k m m=1 k    k (k − m)! (m) αn−l j j=m = Rn (x). Whence the result. xj Djx (Rn (x)) + (j − m)! α0(m) k    k (k − m)! j=m j (j − m)! xj−m Dxn−l+j (Rn (x))  The cases k = 1 and k = 2 are presented in the following corollaries. Corollary 2.3. Choosing k = 1 in Theorems 2.1 and 2.2, by taking A′ (t ) A(t ) = ∞  αn(1) n=0 tn n! αn(1) := αn ; ; αn(0) = δn,0 :=  1, 0, n=0 otherwise, we get the recurrence relation Rn+1 (x) = (x + α0 )Rn (x) + n−1    n l l=0 αn−l Rl (x). On the other hand, 1-times shift operators (or simply the shift operators) are given by Θn− := Φn− = 1 n Dx and Θn+ := (x + α0 ) + n−1  l=0 1 (n − l)! αn−l Dnx −l . Finally, for λn,1 = n, the differential equation is given by (x) Ln,1 (Rn (x)) = nRn (x), where (x) Ln,1 := (x + α0 ) Dx + n−1  l=0 1 (n − l)! αn−l Dxn−l+1 . Note that these results coincide with the results obtained in [8]. Corollary 2.4. Letting k = 2 in Theorems 2.1 and 2.2, by setting A′ (t ) A(t ) = ∞  n=0 αn(1) tn n! and A′′ (t ) A(t ) we get the recurrence relation  (1 ) (2) Rn+2 (x) = x2 + 2α0 x + α0  = ∞  αn(2) n=0 Rn (x) + tn n! n−1     n l=0 l (1) (2)  2xαn−l + αn−l Rl (x). xj−m Djx (Rn (x)) 113 M. Ali Özarslan, B. Yılmaz / Journal of Computational and Applied Mathematics 259 (2014) 108–116 2-times shift operators are given by 1 Θn−(2) := Φn−−1 Φn− = (n − 1)n D2x and n−1        n (2) (2) (1 ) (1) Θn+(2) := x2 + 2α0 x + α0 + 2xαn−l + αn−l Dnx −l . l=0 l Finally, for λn,2 = n2 + 3n, the differential equation is given by (x) Ln,2 (Rn (x)) = n2 + 3n Rn (x),  where (x) Ln,2 :=   4xDx + x  2 D2x  (1) (1) + 4α0 Dx + 2α0 xD2x (2) 2 + α0 Dx + (1) (2) n−1  2αn−l (2Dxn−l+1 + xDnx −l+2 ) + αn−l Dnx −l+2 l=0 (n − l)!  . 3. Applications of main theorems In this section, we apply our results obtained in Section 2 to the two famous representatives of the Appell polynomials, the Hermite and the Bernoulli polynomials. Since the case k = 1 gives the known results for these polynomial sets, we exhibit the case k = 2. 3.1. Hermite polynomials The generating function of the Hermite polynomials is given by the following relation: e2xt − t2 2 = ∞  Hen (x) n=0 tn n! . 2 t If we choose A(t ) = e− 2 we get A′ (t ) A(t ) = −t = ∞  αn(1) n=0 and thus tn n! α0(1) = α2(1) = α3(1) = · · · = 0. α1(1) = −1; On the other hand ′′ A (t ) A(t ) = −1 + t 2 = ∞  αn(2) n=0 so α2(2) = 2; α0(2) = −1, tn n! α1(2) = α3(2) = α4(2) = · · · = 0. Corollary 3.1. Using the above results for k = 2 in Theorems 2.1 and 2.2, the recurrence relation for the Hermite polynomials is Hen+2 (x) = x2 − 1 Hen (x) − 2nxHen−1 (x) + n(n − 1)Hen−2 (x),  the shift operators are Θn−(2) = 1 (n − 1)n  D2x ,   Θn+(2) := x2 − 1 − 2nxDx + n(n − 1)D2x and the fourth order differential equation is given by D2x (x2 − 1 − 2xDx + D2x )Hen (x) = (n + 2)(n + 1)Hen (x). 114 M. Ali Özarslan, B. Yılmaz / Journal of Computational and Applied Mathematics 259 (2014) 108–116 3.2. Bernoulli polynomials The celebrated Bernoulli polynomials are given by the generating relation G(x, t ) = t et −1 ∞  ext = B n ( x) n=0 tn n! |t | < 2π ; (9) and Bernoulli numbers are defined by Bn := Bn (0). First few Bernoulli numbers are B0 = 1, −1 B1 = 2 B2 = , 1 6 B3 = 0, , B4 = −1 (10) 30 and B2k+1 = 0 for (k = 1, 2, . . .). First few Bernoulli polynomials are B0 (x) = 1, B1 (x) = x − 1 2 1 B2 (x) = x2 − x + , 6 B3 (x) = x3 − , 3 2 x2 + 1 2 (11) x. Bernoulli numbers and polynomials play an important role in many areas of mathematics. While the Bernoulli numbers appear in the Maclaurin expansion tan x, cot x, tan hx and coth x, the Bernoulli polynomials appear in the sums of powers of natural numbers: l  kr = Br (l + 1) − Br +1 r +1 k=0 . On the other hand, the Bernoulli numbers appear in the Euler–Maclaurin quadrature formula [27]. The following properties are the characteristic properties of the Bernoulli polynomials: Bn (x) = n    n k k=0 Bk xn−k , Bn (1 − x) = (−1)n Bn (x), n ≥ 0, B′n (x) = nBn−1 (x), Bn (x + 1) − Bn (x) = nxn−1 . (x) Now, we apply the procedure used in Section 2 for the case k = 2 to obtain the differential operator Lm,2 such that (x) Lm,2 (Bm (x)) = m2 + 3m Bm (x).   Taking derivatives with respect to t in (9), we get   ′′ 2xA′ (t ) ∂ 2 G(x, t ) A (t ) 2 = G ( x , t ) + + x ∂t2 A(t ) A(t ) where A(t ) = t . et −1 Therefore we obtain     t et 2et (et − 1 − tet ) e − 1 − tet ∂ 2 G(x, t ) 2 + x = G ( x , t ) − − + 2x ∂t2 et − 1 t (et − 1)2 t (et − 1)       1 − et − 1 1 et tet et 1 2 = G(x, t ) − + x + 2x − 2 − − 1 − et t et − 1 (et − 1)2 t et − 1    t t t e x et e 1 te t t 2 . = G(x, t ) + 2 + 2 − 2x + x − 1 − 2 1 − et t 2 et − 1 t (et − 1) et − 1 t t et − 1 Now, inserting the series relations, we obtain ∞  m=0 Bm+2 (x) tm m! = ∞  Bm (x) t2 m!  − ∞ 1 n=0 k=0 l=0 t k =0 Bk tk k!   ∞ n k 2    n k m=0 + tm k l −1− Bl Bk−l ∞ n 2   n t2 tn n! n=0 k=0 +  2x t k Bk  tn n! + x2 − + ∞ n 2   n t n=0 k=0 ∞ n 2x    n  t n=0 k=0 k k Bk Bk tn n! tn n!  . 115 M. Ali Özarslan, B. Yılmaz / Journal of Computational and Applied Mathematics 259 (2014) 108–116 Using (10) and (11), then regulating the series and comparing the coefficients of relation:  Bm+2 (x) = x2 − x − 2 m+2 − 2 m+1 + 7 6     m + 2 n + 3 −2 Bk − n+3 k n=0 k=0 × m−1 n+2 (m + 1)(m + 2) + 2(1 − x) Bm−k−1 (x) = (m − k − 1)! k+1 Dx Bm (x) m! Bm−n−1 (x) = (m − n − 1)! n+1 Dx Bm (x) m! we have the following recurrence m! Bm−k−1 (x)Bk+2 ( m − k − 1)!(k + 2)! k=0     k   m+2 n+3 k  k l=0 k l Bl Bk−l    m + 1   n + 2  Bk Bm−n−1 (x) n=0 k=0 Since m! m−1 Bm (x) − m−1 n+3 Bm−n−1 (x) tm n+2 m+1 k . and the increasing operator can be written as +(2) Θm 2 2 = x −x− × k   l m+2 − 2 m+1 + 7 6 m−1 −  k=0 m−1 n+3 Bk+2 (k + 2)! Dkx+1 −2  n=0 k=0  Bk (n + 3 − k)!k! m −1  n+2  (m − n − 1)! Bk Bl Bk−l Dxn+1 + 2(1 − x) Dnx +1 . (m + 2 − k)!k! ( n + 2 − k )! k ! n=0 k=0  −  k   n+3 l=0 k On the other hand, noting the fact that the two times lowering operators for all Appell polynomials is Θn−(2) = 1 (n − 1)n D2x by using the generalized factorization method with k = 2 −(2) Θm+2 Θm+(2) Bm (x) = Bm (x) we get after some series manipulations that the differential equation for the Bernoulli polynomials for the case k = 2 can be written as (x) Lm,2 (Bm (x)) = m2 + 3m Bm (x)  where  (x) Lm,2 := x2 Dx + 2xDx − xD2x + 2 − − k   l=0 n+3 k k   l 2 m+2 D2x + 7 6 m−1 D2x −  k=0 Bk+2 (k + 2)! m−1 n+3 Dkx+3 − 2  n=0 k=0 Bk (n + 3 − k)!k! Dnx +3  m −1  n+2  (m − n − 1)! Bk Bl Bk−l Dnx +3 + 2(1 − x) Dn+3 . (m + 2 − k)!k! ( n + 2 − k)!k! x n=0 k=0 References [1] G. Dattoli, Hermite–Bessel, Laguerre–Bessel functions: a by-product of the monomiality principle, in: D. Cocolicchio, G. Dattoli, H.M. Srivastava (Eds.), Advanced Special Functions and Applications, Proceedings of the Melfi School on Advanced Topics in Mathematics and Physics; Melfi, 9–12 May 1999, Aracne Editrice, Rome, 2000, pp. 147–164. [2] Y.B. Cheikh, Some results on quasi-monomiality, Appl. Math. Comput. 141 (2003) 63–76. [3] M.G. Bin-Saad, Associated Laguerre–Konhauser polynomials, quasi-monomiality and operational identities, J. Math. Anal. Appl. 324 (2) (2006) 1438–1448. [4] G. Bretti, P.E. Ricci, Multidimensional extension of the Bernoulli and Appell polynomials, Taiwanese J. Math. 8 (3) (2004) 415–428. [5] C. Cesarano, Monomiality principle and Legendre polynomials, in: G. Dattoli, H.M. Srivastava, C. Cesarano (Eds.), Advanced Special Functions and Integration Methods Y. Ben Cheikh/Appl. Math. Comput. 141 (2003) 63–76, 75, Proceedings of the Melfi School on Advanced Topics in Mathematics and Physics; Melfi, 18–23 June 2000, Aracne Editrice, Rome, 2001, pp. 83–95. [6] G. Dattoli, H.M. Srivastava, C. Cesarano, The Laguerre and Legendre polynomials from an operational point of view, Appl. Math. Comput. 124 (2001) 117–127. [7] G. Dattoli, A. Torre, G. Mazzacurati, Quasi-monomials and isospectral problems, Nuovo Cimento B 112 (1997) 133–138. [8] M.X. He, P.E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2002) 231–237. [9] H.M. Srivastava, Y.B. Cheikh, Orthogonality of some polynomial sets via quasi-monomiality, Appl. Math. Comput. 141 (2–3) (2003) 415–425. [10] H.M. Srivastava, Y.B. Cheikh, Orthogonality of some polynomial set via quasi-monomiality, Appl. Math. Comput. 141 (2003) 415–425. 116 M. Ali Özarslan, B. Yılmaz / Journal of Computational and Applied Mathematics 259 (2014) 108–116 [11] P. Appell, Sur une classe de polynomes, Ann. Sci. Éc. Norm. Supér. 9 (2) (1880) 119–144. [12] T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, London, Paris, 1978. [13] H. Bateman, A. ErdPelyi, Higher Trascendental Functions. The Gamma Function. The Hypergeometric Function. Legendre Functions, Vol. 1, McGrawHill, New York, 1953. [14] T.M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951) 161–167. [15] Q.-M. Luo, H.M. Srivastava, q-extensions of some relationships between the Bernoulli and Euler polynomials, Taiwanese J. Math. 15 (2011) 241–257. [16] H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000) 77–84. [17] H.M. Srivastava, H.L. Manocha, A Treatise On Generating Functions, Wiley, New York, Chichester, Brisbane, Toronto, 1984. [18] A. Erdélyi, A. Magnus, W. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vols. I and II, McGraw-Hill, New York, Toronto, London, 1953. [19] H.W. Gould, A.T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J. 29 (1962) 51–63. [20] T. Kim, S.-H. Rim, Y. Simsek, D. Kim, On the analogs of Bernoulli and Euler numbers, related identities and zeta and L-functions, J. Korean Math. Soc. 45 (2008) 435–453. [21] Q.-M. Luo, H.M. Srivastava, Some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials, J. Math. Anal. Appl. 308 (2005) 290–302. [22] H.M. Srivastava, M. Garg, S. Choudhary, A new generalization of the Bernoulli and related polynomials, Russ. J. Math. Phys. 17 (2010) 251–261. [23] H. Ozden, Y. Simsek, A new extension of q-Euler numbers and polynomials related to their interpolation functions, Appl. Math. Lett. 21 (2008) 934–939. [24] Q.-M. Luo, Extension for the Genocchi polynomials and its Fourier expansions and integral representations, Osaka J. Math. 48 (2011) 291–309. [25] Q.-M. Luo, q-extensions for the Apostol–Genocchi polynomials, Gen. Math. 17 (2009) 113–125. [26] Q.-M. Luo, H.M. Srivastava, Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011) 5702–5728. [27] J. Stoer, Introduzione all’Analisi Numerica, Zanichelli, Bologna, 1972.