Some New Identities Involving Sheffer-Appell Polynomial Sequences Via Matrix Approach

Home , Appell sequence, Sheffer sequence

SOME NEW IDENTITIES INVOLVING SHEFFER-APPELL POLYNOMIAL SEQUENCES VIA MATRIX APPROACH

MOHD SHADAB, FRANCISCO MARCELLAN´ AND SAIMA JABEE∗

Abstract. In this contribution some new identities involving Sheﬀer-Appell polynomial sequences using generalized Pascal functional and Wronskian matrices are deduced. As a direct application of them, identities involving families of polynomials as Euler, Bernoulli, Miller-Lee and Apostol-Euler polynomials, among others, are given.

1. introduction Sequences of polynomials play an important role in many problems of pure and applied mathematics in the framework of approximation theory, statistics, combinatorics and classical analysis (see, for example, [19, 22–25]). The sequence of Sheﬀer polynomials constitutes one of the most important family of polynomial sequences. A polynomial sequence {sn(x)}n≥0 is said to be a Sheﬀer polynomial sequence [6, 9, 24, 27] if its generating function has the following form: ∞ X yn A(y)exH(y) = s (x) , (1.1) n n! n=0 where

A(y) = A0 + A1y + ··· , and 2 H(y) = H1y + H2y + ··· , with A0 6= 0 and H1 6= 0.

Let us recall an alternative definition of the Sheffer polynomial sequences [24, Pg. 17]. P∞ yn P∞ yn Indeed, let h(y) = n=1 hn n! , h1 6= 0, and l(y) = n=0 ln n! , l0 6= 0, be, respectively, delta series and invertible series with complex coefficients. Then there exists a unique sequence of Sheffer polynomials {sn(x)}n≥0 satisfying the orthogonality conditions k hl(y)h(y) |sn(x)i = n!δn,k ∀ n, k = 0, (1.2) where δn,k is the Kronecker delta. Notice that the above orthogonality is defined as P∞ an n follows. Given a formal power series f(x) = n=0 n! x we can introduce a linear func- n tional in the linear space of polynomials associated with f such that hf(x)|x i = an, n ≥ 0, and extended by linearity to every polynomial. See [24, p.6]. The polynomial sequence {sn(x)}n≥0 is said to be polynomial Sheffer sequence for the pair (l(y), h(y)).

2010 Mathematics Subject Classiﬁcation. 15A15, 15A24, 33C45. Key words and phrases. Sheﬀer-Appell polynomial sequence; Generalized Pascal functional; Wron- skian matrices; Identities; Orthogonal polynomials. *Corresponding author. 1 2 M. SHADAB, F. MARCELLAN´ AND SAIMA JABEE

Notice that an algebraic approach to Sheﬀer polynomial sequences has been done in [8]. On the other hand, a perspective about Sheﬀer polynomials and the monomiality principle by using algebraic methods appears in [9].

Roman [24, p. 18, Theorem 2.3.4] introduced the exponential generating function of sn(x) as follows

∞ n 1 ˜ X y exh(y) = s (x) . (1.3) ˜ n n! l(h(y)) n=0 where h˜ is the compositional inverse of h.

The Sheﬀer sequence for the pair (l(y), y) is called an Appell sequence for l(y). In fact, Roman [24] characterized Appell sequences as follows

{αn(x)}n≥0 is an Appell polynomial sequence if either d α (x) = nα (x), n ∈ , dx n n−1 N or if there exists an exponential generating function of the form (see also the recent works [20, 28]) ∞ X yn A(y)exy = α (x) , (1.4) n n! n=0 where N denotes the set of positive integer numbers and 1 A(y) = . l(y) We also note that for H(y) = y, the generating function (1.1) of the Sheﬀer polynomials sn(x) reduces to the generating function (1.4) of the Appell polynomials αn(x).

A determinantal approach to Appell polynomials has been given in [7]. In [13] (see also [26]) He and Ricci derived some recurrence relations and differential equations for the Appell polynomial sequence. Further, in [32] (see also [1]) Youn and Yang obtained some identities and differential equation for the Sheffer polynomial sequence by using matrix algebra.

Now, in order to recall the deﬁnition of the generalized Pascal functional matrix of an analytic function (see [30]), let ( ∞ ) X yr F = h(y) = α , α ∈ . r r! r C r=0

Then the generalized Pascal functional matrix [Pn(h(y))], which is a lower triangular matrix of order (n + 1) × (n + 1) for h(y) ∈ F, is deﬁned by  i  h(i−j)(y), i j,  j = Pn[h(y)]ij = (1.5)   0, otherwise, SOME NEW IDENTITIES INVOLVING SHEFFER-APPELL POLYNOMIAL SEQUENCES 3 for all i, j = 0, 1, 2, ··· , n. Here h(i)(y) denotes the ith order derivative of h(y).

We next recall the nth order Wronskian matrix of analytic functions h1(y), h2(y), ··· , hm(y), with size (n + 1) × m, as follows:   h1(y) h2(y) h3(y) ··· hm(y)  0 0 0 0   h1(y) h2(y) h3(y) ··· hm(y)  W [h (y), h (y), ··· , h (y)] =   . (1.6) n 1 2 m  ......   . . . . .   (n) (n) (n) (n)  h1 (y) h2 (y) h3 (y) ··· hm (y) Next we summarize some properties and relations between the Wronskian matrices and the generalized Pascal functional matrices since they constitute a basic tool of our work (see, for example, [31, 32]).

Property I. For h(y), l(y) ∈ C[y], Pn[h(y)] and Wn[h(y)] satisfy that is,

Pn[uh(y) + vl(y)] = uPn[h(y)] + vPn[l(y)] and

Wn[uh(y) + vl(y)] = uWn[h(y)] + vWn[l(y)], where u, v ∈ C.

Property II. For h(y), l(y) ∈ C[y],

Pn[h(y)l(y)] = Pn[h(y)]Pn[l(y)] = Pn[l(y)]Pn[h(y)]. Property III. For h(y), l(y) ∈ C[y],

Wn[h(y)l(y)] = Pn[h(y)]Wn[l(y)] = Pn[l(y)]Wn[h(y)]. 0 Property IV. For h(y), l(y) ∈ C[y], with h(0) = 0 and h (0) 6= 0, 2 n −1 Wn[l(h(y))] y=0 = Wn 1, h(y), h (y), ··· , h (y) y=0 Ωn Wn[l(y)] y=0, where

Ωn = diag[0!, 1!, 2!, ··· , n!]. In [15] the authors introduced the Sheffer-Appell polynomials as a discrete Appell convolution of Sheffer polynomials. Its generating function, series definition as well as a determinantal definition are deduced. These polynomials are strongly related to the monomiality principle and some of their properties are presented. Results for the Sheffer-Bernoulli and Sheffer-Euler polynomials are obtained. In particular, differential equations satisfied by these polynomials are given.

The Sheffer-Appell polynomial sequences are umbral composition ([24], pg. 41) of Appell and Sheffer polynomial sequences. Hence they are particular Sheffer sequences. The generating function of the Sheffer-Appell polynomial sequence {sAn(x)}n≥0 is defined as ∞ n 1 −1 X y exh (y) = A (x) , (1.7) l(h−1(y))l(y) s n n! n=0 4 M. SHADAB, F. MARCELLAN´ AND SAIMA JABEE

1 xh−1(y) where l(h−1(y))l(y) e is analytic. Then by using Taylor’s theorem, we obtain

k d 1 xh−1(y) sAk(x) = k −1 e , k ≥ 0. (1.8) dy l(h (y))l(y) y=0

For the Sheﬀer-Appell polynomial sequence {sAn(x)}n≥0 associated to the pair (l(y), h(y)) we introduce the vector

T sA~ n(x) = [sA0(x), sA1(x),..., sAn(x)] , (1.9) which can also be expressed as ~ T 1 xh−1(y) sAn(x) = [sA0(x), sA1(x),..., sAn(x)] = Wn −1 e . (1.10) l(h (y))l(y) y=0 As an auxiliary result we deduce an expression for the Wronskian matrix of the vector sA~ n(x).

Lemma 1.1. Let {sAn(x)}n≥0 be the Sheﬀer-Appell polynomial sequence associated with the pair (l(y), h(y)). Then ~ T −1 Wn[sAn(x)] Ωn −1 −1 2 −1 n = Wn[1, (h (y)), (h (y)) , ··· , (h (y)) ] y=0

1 1 · Ω−1P P P [exy] . (1.11) n n l(y) n l(h(y)) n y=0 y=0 y=0

Proof. Let us begin with (1.10), that is, 1 −1 A~ (x) = W exh (y) . (1.12) s n n l(h−1(y))l(y) y=0 Applying Property IV in (1.12), we get

1 A~ (x) = W [1, (h−1(y)), (h−1(y))2, ··· , (h−1(y))n] Ω−1W exy (1.13). s n n n n l(y)l(h(y)) y=0 y=0 Taking into account

xy 2 n T Wn[e ] = [1, x, x , ··· , x ] , (1.14) y=0 then (1.13) becomes

~ −1 −1 2 −1 n sAn(x) = Wn[1, (h (y)), (h (y)) , ··· , (h (y)) ] y=0

1 1 · Ω−1P P [1, x, x2, ··· , xn]T . (1.15) n n l(y) n l(h(y)) y=0 y=0 SOME NEW IDENTITIES INVOLVING SHEFFER-APPELL POLYNOMIAL SEQUENCES 5

Now, by taking the kth order derivative with respect to x in both hand sides of (1.15) and dividing the resulting equation by k!, we obtain 1 [ A(k)(x), A(k)(x), ··· , A(k)(x)]T k! s 0 s 1 s n

−1 −1 2 −1 n = Wn[1, (h (y)), (h (y)) , ··· , (h (y)) ] y=0

1 1 · Ω−1P P n n l(y) n l(h(y)) y=0 y=0 k + 1 k + 2 n T · 0, ··· , 0, 1, x, x2, ··· , xn−k . (1.16) k k k Hence, the right-hand side and left-hand side of (1.16) are the kth columns of

1 1 W [1, (h−1(y)), (h−1(y))2, ··· , (h−1(y))n] Ω−1P P P [exy] n n n l(y) n l(h(y)) n y=0 y=0 y=0 y=0 and T −1 Wn[sA0(x), sA1(x), ··· , sAn(x)] Ω , respectively. Thus, the statement of Lemma 1.1 follows. Once we have this basic background, we describe the structure of the paper. In Section 2, we derive some identities involving Sheﬀer-Appell polynomials. A relation between two diﬀerent sequences of such polynomials is also deduced in Theorem 6. In Section 3 we present some applications of the preceding results to particular families as Laguerre type Appell polynomials, Apostol-Euler-Appell polynomials and Miller-Lee type Appell polynomials.

2. Some identities involving Sheffer-Appell polynomial sequences via matrix approach Identities involving orthogonal polynomials are obtained in the literature by using diﬀerent approaches (see, e.g. [5, 11, 12, 14]). In this section, we derive some results for Sheﬀer-Appell polynomial sequences by using the generalized Pascal functional and Wronskian matrices.

First, we obtain a representation of the polynomial of degree n + 1 in terms of the previous ones.

Theorem 1. The Sheﬀer-Appell polynomial sequence sAn(x) ∼ (l(y), h(y)) satisﬁes the following relation n (k) n X sAn (x) X n a A (x) = (b x − b − c ) − a A (x). (2.1) 0s n+1 k! k k+1 k k − 1 n−k+1s k k=0 k=1 Here

0 −1 −1 (k) ak = h (h (y))l(h (y)) , k = 0, y=0 6 M. SHADAB, F. MARCELLAN´ AND SAIMA JABEE

0 (k) bk = l (y) , k = 0, y=0 and 0 0 (k) h (y)l (h(y))l(y) ck = , k = 0. l(h(y)) y=0

Proof. Let consider

" xh−1(y) !# 0 −1 −1 d e Wn h (h (y))l(h (y)) −1 . (2.2) dy l(h (y))l(y) y=0 On one hand, applying property III, we get

" xh−1(y) !# 0 −1 −1 d e Pn h (h (y))l(h (y)) Wn −1 y=0 dy l(h (y))l(y) y=0 or, equivalently,   a0 0 0 ... 0   sA1(x) a a 0 ... 0   0 1   sA2(x)   2    a2 a1 a0 ... 0   sA3(x)  =  1    . (2.3)  . . . . .   .   ......   .    n sAn+1(x) an 1 an−1 ...... a0 On the other hand, we can write (2.2) as

" 0 −1 0 −1 xh−1(y) # −1 h (h (y))l (y)l(h (y)) 0 −1 e Wn xl(h (y)) − − l (h (y)) −1 . (2.4) l(y) l(h (y))l(y) y=0 Applying property IV in (2.4), we get

−1 −1 2 −1 n −1 = Wn 1, h (y), (h (y)) ,..., (h (y)) Ωn y=0 0 0 xy h (y)l(y)l (h(y)) 0 e Wn xl(y) − − l (y) (2.5) l(h(y)) l(h(y))l(y) y=0 According to property III, (2.5) reads −1 −1 2 −1 n −1 1 1 = Wn 1, h (y), (h (y)) ,..., (h (y)) Ωn Pn Pn y=0 l(y) y=0 l(h(y)) y=0 0 0 xy h (y)l(y)l (h(y)) 0 .Pn [e ] Wn xl(y) − − l (y) . (2.6) y=0 l(h(y)) y=0 From Lemma 1.1 we get T −1 = Wn [sA0(x), sA1(x), sA2(x),..., sAn(x)] Ωn " 0 0 # h (y)l(y)l (h(y)) 0 xWn[l(y)] − Wn − Wn[l (y)] y=0 l(h(y)) y=0 y=0 SOME NEW IDENTITIES INVOLVING SHEFFER-APPELL POLYNOMIAL SEQUENCES 7 or, equivalently,   sA0(x) 0 0 ... 0 0  xb − b − c   sA1(x)  0 1 0 sA1(x) 0 ... 0   1!   xb1 − b2 − c1   0 00    sA2(x) sA2 (x) = sA2(x) ... 0   xb2 − b3 − c2  . (2.7)  1! 2!   .   . . . . .   .   ......      xb − b − c  0 00 (n)  n n+1 n sAn(x) sAn(x) sAn (x) sAn(x) 1! 2! ... n! Finally, identifying the nth rows of (2.3) and (2.7), the statement follows. Next, we obtain an interesting relation satisﬁed by the polynomial of degree n and their derivatives in terms of the polynomials of degree at most n − 1.

Theorem 2. Let sAn(x) ∼ (l(y), h(y)) be the Sheﬀer-Appell polynomial sequence. Then, n n (k) X n − 1 X sAn (x) n a A (x) = (xb + c + d ) , k − 1 n−ks k k k k k! k=1 k=0 (2.8) where 0 −1 (k) ak = h (h (y)) , k = 0, y=0

(k) bk = (h(y)) , k = 0, y=0 0 0 (k) −h(y)l (h(y)))h (y) ck = , k = 0, l(h(y)) y=0 and 0 (k) −l (y)h(y) dk = , k = 0. l(y) y=0 Proof. From " xh−1(y) !# 0 −1 d e Wn yh (h (y)) −1 (2.9) dy l(h (y))l(y) y=0 and applying property III in (2.9), we get " xh−1(y) !# 0 −1 d e Pn [y] Pn h (h (y) Wn −1 y=0 y=0 dy l(h (y))l(y) y=0 or, equivalently, 0 0 0 ... 0 0 0     1 0 0 ... 0 0 0 a0 0 0 ... 0 sA1(x)   0 2 0 ... 0 0 0 a1 a0 0 ... 0   sA2(x)     2    0 0 3 ... 0 0 0 a2 a1 a0 ... 0   .  =    1   .  . ......   ......    ......   . . . . .   sAn(x)    n n 0 0 0 . . . n − 1 0 0 an 1 an−1 2 an−2 . . . a0 sAn+1(x) 0 0 0 ... 0 n 0 (2.10) 8 M. SHADAB, F. MARCELLAN´ AND SAIMA JABEE

On the other hand, we can rewrite (2.9) as " 0 −1 0 −1 0 −1 −1 xh−1(y) # −1 h (h (y))l (y)h(h (y)) l (h (y))h(h (y)) e Wn xh(h (y)) − − −1 −1 . l(y) l(h (y)) l(h (y))l(y) y=0 (2.11) From property IV we deduce

−1 −1 2 −1 n −1 = Wn[1, (h (y)), (h (y)) ,..., (h (y)) ] Ωn y=0 0 0 0 xy h (y)l (h(y))h(y) l (y)h(y) e .Wn xh(y) − − . (2.12) l(h(y)) l(y) l(y)l(h(y)) y=0 Next, using property III in (2.12), we get −1 −1 2 −1 n 1 1 = Wn[1, (h (y)), (h (y)) ,..., (h (y)) ] Pn Pn y=0 l(y) y=0 l(h(y)) y=0 0 0 0 xy −1 h (y)l (h(y))h(y) l (y)h(y) .Pn [e ] Ωn Wn xh(y) − − . (2.13) y=0 l(h(y)) l(y) y=0 As a consequence of Lemma 1.1, T −1 = Wn [sA0(x), sA1(x), sA2(x),..., sAn(x)] Ωn " 0 0 0 # h (y)l (h(y))h(y) l (y)h(y) . xWn [h(y)] + Wn − + Wn − y=0 l(h(y)) y=0 l(y) y=0 or, equivalently, T −1 = Wn [sA0(x), sA1(x), sA2(x),..., sAn(x)] Ωn T .[xb0 + c0 + d0, xb1 + c1 + d1, . . . , xbn + cn + dn] . (2.14) Equating the nth rows of (2.10) and (2.14), our statement follows. On the other hand, we get an algebraic relation for a Sheﬀer-Appell polynomial of degree n and their derivatives.

Theorem 3. The Sheﬀer-Appell polynomial sequence sAn(x) ∼ (l(y), h(y)) satisﬁes n X xk a = (−1)k A(k)(x) , (2.15) n s n k! k=0 where (k) 1 ak = −1 , k = 0. l(y)l(h (y)) y=0

Proof. Let consider 1 Wn −1 . (2.16) l(h (y))l(y) y=0 On one hand, from (1.6), we get 1 T Wn −1 = [a0, a1, a2, . . . , an] . (2.17) l(h (y))l(y) y=0 SOME NEW IDENTITIES INVOLVING SHEFFER-APPELL POLYNOMIAL SEQUENCES 9

On the other hand, (2.16) can be written as

" xh−1(y) # e −xh−1(y) Wn −1 e . (2.18) l(h (y))l(y) y=0 By using property IV in (2.18), we get xy −1 −1 2 −1 n −1 e −xy = Wn 1, h (y), (h (y)) ,..., (h (y)) Ωn Wn e . (2.19) y=0 l(y)l(h(y)) y=0 Thus, according to property III, (2.19) becomes −1 −1 2 −1 n −1 1 = Wn 1, h (y), (h (y)) ,..., (h (y)) Ωn Pn y=0 l(y) y=0 1 xy −xy .Pn Pn [e ] Wn e , (2.20) l(h(y)) y=0 y=0 y=0 and applying Lemma 1.1, we obtain

T −1 2 nT = Wn [sA0(x), sA1(x), sA2(x),..., sAn(x)] Ωn 1, −x, (−x) ,..., (−x) . (2.21) Identifying the nth rows of (2.17) and (2.21), we get the desired result. Next, we obtain a representation of the Sheﬀer-Appell polynomial of degree n − 1 in terms of the polynomial of degree n and their derivatives.

Theorem 4. Let sAn(x) ∼ (l(y), h(y)) be the Sheﬀer-Appell polynomial sequence. Then, n (k) X sAn (x) a = n A (x), (2.22) k k! s n−1 k=1 where (k) ak = h (0), 1 ≤ k ≤ n. Proof. Let consider

" xh−1(y) !# e Wn y −1 . (2.23) l(h (y))l(y) y=0 On one hand, from property III, (2.23) reads

" xh−1(y) !# " xh−1(y) # e e Wn y −1 = Pn[y] Wn −1 l(h (y))l(y) y=0 y=0 l(h (y))l(y) y=0

0 0 0 ... 0 0 0   1 0 0 ... 0 0 0 sA0(x)   0 2 0 ... 0 0 0  sA1(x)      0 0 3 ... 0 0 0  sA2(x)  = ......   .  . (2.24) ......   .  ......   .  0 0 0 ... 0 0 0  A (x)   s n−1  0 0 0 . . . n − 1 0 0 sAn(x) 0 0 0 ... 0 n 0 10 M. SHADAB, F. MARCELLAN´ AND SAIMA JABEE

According to property IV (2.24) becomes " xh−1(y) !# e −1 −1 2 −1 n −1 Wn y −1 = Wn[1, (h (y)), (h (y)) ,..., (h (y)) ] Ωn l(h (y))l(y) y=0 y=0 xy e × Wn h(y) . l(y)l(h(y)) y=0 (2.25) By applying property III, it yields " xh−1(y) !# e −1 −1 2 −1 n −1 Wn y −1 = Wn[1, (h (y)), (h (y)) ,..., (h (y)) ] Ωn l(h (y))l(y) y=0 y=0 xy 1 1 .Pn [e ] Pn Pn Wn [h(y)] . y=0 l(y) y=0 l(h(y)) y=0 y=0 (2.26) Finally, as a consequence of Lemma 1.1, we get

T −1 = Wn[sA0(x), sA1(x),..., sAn(x)] Ω Wn [h(y)] y=0 or, equivalently,   sA0(x) 0 0 ... 0 0 a   sA1(x)  1 sA1(x) 0 ... 0   1!  a2  0 00    sA2(x) sA2 (x) = sA2(x) ... 0  a3 . (2.27)  1! 2!   .   . . . . .   .   ......      a  0 00 (n)  n sAn(x) sAn(x) sAn (x) sAn(x) 1! 2! ... n! Equating nth rows of (2.24) and (2.27), we get the statement. Next, a connection between the Sheﬀer-Appell polynomials and the function h is deduced.

Theorem 5. The Sheﬀer-Appell polynomial sequence sAn(x) ∼ (l(y), h(y)) satisﬁes n n X n X (n) xk a A (x) = (h−1)k (0) , (2.28) k n−ks k k! k=0 k=1 where

−1 (k) ak = l(y)l(h (y)) , k = 0. y=0 Proof. Let consider the expression

h xh−1(y)i Wn e (2.29) y=0 or, equivalently, " xh−1(y) # −1 e Wn l(y)l(h (y)) −1 . (2.30) l(y)l(h (y)) y=0 SOME NEW IDENTITIES INVOLVING SHEFFER-APPELL POLYNOMIAL SEQUENCES 11

On one hand, from property III we get " xh−1(y) # " xh−1(y) # −1 e −1 e Wn l(y)l(h (y)) −1 = Pn[l(y)l(h (y))] .Wn −1 l(y)l(h (y)) y=0 y=0 l(y)l(h (y)) y=0     a0 0 0 ... 0 sA0(x) a1 a0 0 ... 0  sA1(x)  2    a2 a1 a0 ... 0  sA2(x) =  1    . (2.31)  ......   .   . . . . .   .  n n an 1 an−1 2 an−2 . . . a0 sAn(x) On the other hand, by applying property IV in (2.30), we deduce " xh−1(y) # −1 e Wn l(y)l(h (y)) −1 l(y)l(h (y)) y=0

−1 −1 2 −1 n −1 xy = Wn[1, (h (y)), (h (y)) ,..., (h (y)) ] Ωn .Wn [e ] , y=0 y=0 or, equivalently, 1 0 0 ... 0   1   (h−1)(1)(0) ((h−1)2)(1)(0) ((h−1)n)(1)(0)   x  0 ...     1! 2! n!  x2 = . . . . .    . (2.32) ......   .     .  (h−1)(n)(0) ((h−1)2)(n)(0) ((h−1)n)(n)(0) xn 0 1! 2! ... n! Equating nth rows of (2.31) and (2.32), we get the result stated in the Theorem 6. Our next step is two find a relation between two different sequences of Sheffer-Appell poynomial sequences. 0 Theorem 6. Let sAn(x) ∼ (l(y), h(y)) and rAn(x) ∼ (h (y), h(y)) be the two Sheffer- Appell polynomial sequences. Then, n X n a A (x) + a A (x) 0s n+1 k − 1 n−k+1s k k=1 n−1 X n = (xb + c + d ) A (x) + (xb + c + d ) A (x), (2.33) 0 0 0 r n k n−k n−k n−k r k k=1 where

−1 (k) ak = l(y)l(h (y)) , k = 0, y=0

0 (k) bk = h (y) , k = 0, y=0 0 −1 0 0 (k) −h (h (y))l (y)h (y) ck = , k = 0, l(y) y=0 and 0 −1 0 (k) −l (h (y))h (y) dk = −1 , k = 0. l(h (y)) y=0 12 M. SHADAB, F. MARCELLAN´ AND SAIMA JABEE

Proof. Let consider

" xh−1(y) !# −1 d e Wn l(h (y))l(y) −1 . (2.34) dy l(h (y))l(y) y=0 On one hand, from property III (2.34) reads as

" xh−1(y) !# d e −1 Pn −1 Wn l(h (y))l(y) dy l(h (y))l(y) y=0 y=0 or, equivalently,   sA1(x) 0 0 ... 0   a0  A (x) A (x) 0 ... 0   s 2 s 1  a1  2     sA3(x) sA2(x) sA1(x) ... 0  a2 =  1    . (2.35)  . . . . .   .   ......   .    n an sAn+1(x) 1 sAn(x) ...... nsA1(x) On the other hand, (2.34) can be written as

" 0 −1 0 0 0 −1 0 xh−1(y) # 0 h (h (y))l (y)h (y) l (h (y))h (y) e Wn xh (y) − − −1 0 −1 0 . (2.36) l(y) l(h (y)) h (h (y))h (y) y=0 By using property III, we get

0 −1 0 0 0 −1 0 " xh−1(y) # 0 h (h (y))l (y)h (y) l (h (y))h (y) e = Pn xh (y) − − −1 Wn 0 −1 0 . l(y) l(h (y)) y=0 h (h (y))h (y) y=0 " 0 −1 0 0 0 −h (h (y))l (y)h (y) = xPn h (y) + Pn y=0 l(y) y=0 0 −1 0 # −l (h (y))h (y) T + Pn −1 [r0(x), r1(x), . . . , rn(x)] l(h (y)) y=0 or, equivalently,   xb0 + c0 + d0 0 0 ... 0   xb1 + c1 + d1 xb0 + c0 + d0 0 ... 0   2 2 2  xb2 + c2 + d2 x b1 + c1 + d1 xb0 + c0 + d0 ... 0  =  1 1 1   . . . . .   ......    n n n xbn + cn + dn x 1 bn−1 + 1 cn−1 + 1 dn−1 ...... xb0 + c0 + d0   rA0(x) rA1(x)   rA2(x) .   . (2.37)  .   .  rAn(x)

th Equating the n rows of (2.35) and (2.37), we get the statement. SOME NEW IDENTITIES INVOLVING SHEFFER-APPELL POLYNOMIAL SEQUENCES 13

3. Examples Examples of Theorem 1. By applying Theorem 1 to Euler type polynomials [21] ey + 1 E (x) , y , n ∼ 2 we have 1 a = b = c = , k 0. k k k 2 = Hence, we get the following expression for the Euler type Appell polynomials EAn(x): n n x X n 1 X n A (x) = − 1 A (x) − A (x). (3.1) E n+1 2 k E n−k 2 k − 1 E k k=0 k=1 If we apply Theorem 1 to the Miller-Lee type polynomials [2] (m) m+1 Gn (x) ∼ (1 − y) , y , we have ak = bk = ck = (−m − 1)k, k = 0. Thus the following expression for the Miller-Lee type Appell polynomials GAn(x) holds n X n A (x) = (x + m − k)(−m − 1) A (x) G n+1 k kG n−k k=0 n X n − (−m − 1) A (x). (3.2) k − 1 n−k+1G k k=1 For the generalized Laguerre polynomials [21] y Lλ(x) (1 − y)−λ−1, , n ∼ y − 1 from the statement of Theorem 1 we get

ak = (−λ − 1)k, k = 0, bk = (λ + 1)k, k = 0, and ck = (λ + 1)(λ + 4)k, k = 0. As a consequence, for the Laguerre type Appell polynomials LAn(x) we get n X n A (x) = A (x)[(λ + 1) x − (λ + 1) L n+1 k L n−k k k+1 k=0 n X n −(λ + 1)(λ + 4) ] − (−λ − 1) A (x). (3.3) k k − 1 n−k+1L k k=1 Let consider the Apostol-Euler type polynomials, see [17, 18] λey + 1 (x; λ) , y . n ∼ y Then, ( λ+1 2 , k = 0, ak = bk = λ 2 , k > 0, 14 M. SHADAB, F. MARCELLAN´ AND SAIMA JABEE and λ c = , k 0,. k 2 = Hence, for the Apostol-Euler-Appell polynomials An(x; λ) the statement of Theorem 1 reads as n X n (λ + 1) A (x; λ) − [(λ + 1)x − λ] A (x; λ) = λ(x − 2) A (x; λ) n+1 n k n−k k=1 n X n −λ A (x; λ). (3.4) k − 1 k k=1 λ −λ−1 y Example of Theorem 2. For the generalized Laguerre polynomials Ln(x) ∼ (1 − y) , y−1 , we have ( −1, k = 1, ak = 0, otherwise, ( 0, k = 0, bk = −(k)!, k > 0,

ck = (λ + 1)[(4)k − (3)k],

dk = (λ + 1)[(1)k − (2)k]. As a consequence, for the Laguerre type Appell polynomials LAn(x) we obtain n X n −n(n − 1) A (x) = A (x)[(λ + 1)[(4) − (3) − (2) + (1) − x(1) ]. L n−1 k L n−k k k k k k k=1 (3.5) Example of Theorem 3. By applying Theorem 3 to Miller-Lee type polynomials given by (m) m+1 Gn (x) ∼ (1 − y) , y , we have ak = (2m + 2)k, k = 0. Hence, for Miller-Lee-Appell polynomials GAn(x) we have the following algebraic relation n X xk (2m + 2) = (−1)k A(k)(x) . (3.6) n G n k! k=0 y y Examples of Theorem 4. For the Bernoulli type polynomials Bn(x) ∼ ey−1 , e − 1 , [21] in Theorem 4 we have ( 0, k = 0, ak = 1, k > 0.

As a consequence, for the Bernoulli type Appell polynomials BAn(x) we get n X n n A (x) = A (x). (3.7) B n−1 k B n−k k=1 λ −λ−1 y For the generalized Laguerre polynomials Ln(x) ∼ (1 − y) , y−1 , we have SOME NEW IDENTITIES INVOLVING SHEFFER-APPELL POLYNOMIAL SEQUENCES 15

( 0, k = 0, ak = −(k)!, k > 0.

Thus, for the Laguerre type Appell polynomials LAn(x) we obtain n X n n A (x) = − A (x)k!. (3.8) L n−1 k L n−k k=1 ey+1 Examples of Theorem 5. For the Euler type polynomials En(x) ∼ 2 , y , we have ( 1, k = 0, ak = 2k+2 4 , k > 0.

Thus, for the Euler type Appell polynomials EAk(x) we get

n n−1 X 1 X n A (x) = ((f −1)k)(n)(0) − (2n−k + 2) A (x). (3.9) E k 4 k E k k=1 k=0

(m) m+1 When we deal with the Miller-Lee type polynomials Gn (x) ∼ (1 − y) , y , we have

ak = (−2m − 2)k, k = 0.

Then, for the Miller-Lee type Appell polynomials GAn(x) n n X n X xk (−2m − 2) A (x) = ((f −1)k)(n)(0). (3.10) k n−kG k k! k=0 k=1 Example of Theorem 6. Applying Theorem 6 to the Miller-Lee type polynomials (m) m+1 Gn (x) ∼ (1 − y) , y , we have

ak = ck = dk = −(m + 1)k!, k = 0, and ( 1, k = 0, bk = 0, k > 0. Hence, n X n x A (x) + (n − k + 1)! A (x) = 2 − A (x) G n+1 k − 1 G k m + 1 r n k=1 n−1 X n +2 (n − k)! A (x). (3.11) k r k k=1

4. Acknowledgements The authors thanks the constructve comments and suggestions by the referees. They have contributed to improve the presentation of this manuscript.The work of the second author (FM) has been supported by Ministerio de Econom´ıa,Industria y Competitividad of Spain, grant MTM2015-65888-C4-2-P. 16 M. SHADAB, F. MARCELLAN´ AND SAIMA JABEE

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Mohd Shadab: Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (A Central University), New Delhi 110025, India. Email address: [email protected]

Francisco Marcellan:´ Departamento de Matematicas,´ Universidad Carlos III de Madrid, Spain. Email address: [email protected]

Saima Jabee: Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (A Central University), New Delhi 110025, India. Email address: [email protected]