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

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Some New Identities Involving Sheffer-Appell Polynomial Sequences Via Matrix Approach 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 polyno- mial sequences using generalized Pascal functional and Wronskian matrices are de- duced. 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 fsn(x)gn≥0 is said to be a Sheffer polynomial sequence [6, 9, 24, 27] if its generating function has the following form: 1 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]. P1 yn P1 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 fsn(x)gn≥0 satisfying the orthogonality conditions k hl(y)h(y) jsn(x)i = n!δn;k 8 n; k = 0; (1.2) where δn;k is the Kronecker delta. Notice that the above orthogonality is defined as P1 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)jx i = an; n ≥ 0; and extended by linearity to every polynomial. See [24, p.6]. The polynomial sequence fsn(x)gn≥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 1 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 fαn(x)gn≥0 is an Appell polynomial sequence if either d α (x) = nα (x); n 2 ; dx n n−1 N or if there exists an exponential generating function of the form (see also the recent works [20, 28]) 1 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 ( 1 ) X yr F = h(y) = α ; α 2 : 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) 2 F, is defined by 8 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: 2 3 h1(y) h2(y) h3(y) ··· hm(y) 6 0 0 0 0 7 6 h1(y) h2(y) h3(y) ··· hm(y) 7 W [h (y); h (y); ··· ; h (y)] = 6 7 : (1.6) n 1 2 m 6 . .. 7 6 . 7 4 (n) (n) (n) (n) 5 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) 2 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 2 C. Property II. For h(y); l(y) 2 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) 2 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) 2 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 fsAn(x)gn≥0 is de- fined as 1 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 fsAn(x)gn≥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 fsAn(x)gn≥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.
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