mathematics Article Finding Determinant Forms of Certain Hybrid Sheffer Sequences Monairah Alansari 1, Mumtaz Riyasat 2,* , Subuhi Khan 2 and Kaleem Raza Kazmi 2,3 1 Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia; [email protected] 2 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India; [email protected] or [email protected] 3 Department of Mathematics, Faculty of Science & Arts-Rabigh, King Abdulaziz University, Jeddah 21589, Saudi Arabia; [email protected] or [email protected] * Correspondence: [email protected] or [email protected] Received: 12 October 2019; Accepted: 11 November 2019; Published: 14 November 2019 Abstract: In this article, the integral transform is used to introduce a new family of extended hybrid Sheffer sequences via generating functions and operational rules. The determinant forms and other properties of these sequences are established using a matrix approach. The corresponding results for the extended hybrid Appell sequences are also obtained. Certain examples in terms of the members of the extended hybrid Sheffer and Appell sequences are framed. By employing operational rules, the identities involving the Lah, Stirling and Pascal matrices are derived for the aforementioned sequences. Keywords: Sheffer sequences; extended hybrid Sheffer sequences; fractional operators; operational rules; Riordan matrix 1. Introduction Fractional calculus is a branch of mathematics that deals with the real or complex number powers of the differential operator. It is shown in [1] that the exploitation of integral transforms with special polynomials is an effective way to accord with fractional derivatives. Riemann and Liouville [2,3] were the first to use the integral transforms to deal with fractional derivatives. Since differentiation and integration are usually regarded as discrete operations, therefore it is useful to evaluate a fractional derivative. We recall the following definitions: Definition 1. The Euler G-function [4] is given by ¥ Z G(x) = tx−1e−xdt, Re(x) > 0. (1) 0 Definition 2. The Euler’s integral ([5], p. 218) is given by (see also [1]) 1 Z ¥ a−n = e−attn−1dt, minfRe(n), Re(a)g > 0. (2) G(n) 0 Let K be a field of characteristic zero and F be the set of all formal power series in t over K. Let ¥ k f (t) = ∑ akt , k=0 Mathematics 2019, 7, 1105; doi:10.3390/math7111105 www.mdpi.com/journal/mathematics Mathematics 2019, 7, 1105 2 of 16 where ak 2 K for all k 2 N := f0, 1, 2, ...g. The order o( f (t)) of a power series f (t) is the smallest integer k for which the coefficient of tk does not vanish. The series f (t) has a multiplicative inverse, ( )−1 1 ( ( )) = ( ) denoted by f t or f (t) , if and only if o f t 0, then f t is called an invertible series. The series f (t) has a compositional inverse f (t) such that f ( f (t)) = f ( f (t)) = t, if and only if o( f (t)) = 1, then f (t) with o( f (t)) = 1 is called a delta series. Definition 3. An invertible series g(t) and delta series f (t) with ¥ tn ¥ tn f (t) = ∑ fn , f0 = 0; f1 6= 0, g(t) = ∑ gn , g0 6= 0, (3) n=0 n! n=0 n! together form a Sheffer sequence (sn(x))n2N for the pair (g(t), f (t)). The generating function for the Sheffer sequences sn(x) [6] is given by ¥ n 1 x( f (t)) t e = ∑ sn(x) , (4) g( f (t)) n=0 n! which for (1, f (t)) reduces to the associated Sheffer sequence pn(x). A polynomial sequence pn(x) which is binomial type ([7], p. 96) is related to a Sheffer sequence sn(x) by the functional equation: n n sn(x + y) = ∑ sk(x)pn−k(y), (5) k=0 k for all n ≥ 0 and y 2 K, where n being the degree of polynomial and K a field of characteristic zero. Let (sn(x))n2N be a Sheffer sequence for (g(t), f (t)) and suppose n n x = ∑ bn,ksk(x), k=0 then the Sheffer sequence sn(x) can be expressed by the following determinant form: 1 s0(x) = , b0,0 1 x x2 ··· xn−1 xn b0,0 b1,0 b2,0 ··· bn−1,0 bn,0 n (−1) 0 b1,1 b2,1 ··· bn−1,1 bn,1 , (6) sn(x) = b0,0 b ... bn,n 1,1 0 0 b ··· b − b 2,2 n 1,2 n,2 ··· ... .. ... ··· .. 0 0 0 ··· bn−1,n−1 bn,n−1 where bn,k is the (n, k) entry of the Riordan matrix (g(t), f (t)) which defines an infinite, lower triangular array (bn,k)0≤k≤n<¥ according to the following rule: tn ( f (t))k bn,k = g(t) , (7) cn ck ( )( ( ))k where the functions g t f t are called the column generating functions of the Riordan matrix. ck Mathematics 2019, 7, 1105 3 of 16 A vast literature associated with the matrix and other approaches to several special polynomials and corresponding hybrid forms can be found, see [8–17]. These matrix forms helps in solving various algorithms and in finding the solution of numerical and a general linear interpolation problems. The Appell sequence An(x) [6] for (g(t), t) are defined by 1 ¥ tn ext = A (x) , (8) ( ) ∑ n g t n=0 n! which also satisfies the functional equation n n n−k An(x + w) = ∑ Ak(x)w . (9) k=0 k The multi-variable forms of special polynomials are studied in a different way via operational techniques. These may also help in solving problems in classical and quantum mechanics associated with special functions. We recall the following definitions: (r) Definition 4. The 2-variable truncated exponential polynomials (2VTEP) (of order r) en (x, y) are defined by the following generating function, series expansion and operational rule ([18], p. 174 (30)): p ¥ n yt t e J0(2t −x) = ∑ Sn(x, y) , (10) n=0 n! where J0(x) are the regular cylindrical Bessel function, of zero-th order n [ r ] k n−rk (r) y x e (x, y) = n! , (11) n ∑ ( − ) k=0 n rk ! (r) ¶ e (x, y) = exp yD yDr fxng D := . (12) n y x y ¶y (r) Using operational techniques and by convoluting the 2VTEP en (x, y) with Sheffer sequences [19], a class of hybrid Sheffer sequences namely the 2-variable truncated exponential-Sheffer sequences (r) (2VTESS) esn (x, y) are introduced. (r) Definition 5. The exponential generating function and operational rule for the 2VTESS esn (x, y) are given by x f (t) ¥ n 1 e (r) t = s (x, y) , (13) r ∑ e n g( f (t)) (1 − y( f (t)) ) n=0 n! (r) r esn (x, y) = exp yDyyDx fsn(x)g. (14) Remark 1. Taking f (t) = f (t) = t (r) in 2VTESS esn (x, y), we find the 2-variable truncated exponential-Appell sequences (2VTEAS) (r) e An (x, y) [20], which are defined by the following generating function and operational rule: xt ¥ n 1 e (r) t = A (x, y) , (15) ( ) ( − r) ∑ e n g t 1 yt n=0 n! Mathematics 2019, 7, 1105 4 of 16 (r) r e An (x, y) = exp yDyyDx fAn(x)g. (16) (r) The Equation (12) gives the operational rule to introduce the 2VTEP en (x, y) while (14) and (16) (r) define the operational connections between the 2VTESS esn (x, y) and the Sheffer sequences and (r) 2VTEAS e An (x, y) and the Sheffer sequences, obtained by utilizing Equation (12). The Euler’s integral forms the basis of new generalizations of special polynomials. Additionally, the combination of the properties of exponential operators with suitable integral representations yields an efficient way of treating fractional operators. Dattolli et al. [1,21,22] used the Euler’s integral to find the operational definitions and the generating relations for the generalized and new forms of special polynomials. In this article, the exponential operational rule and generating function of the truncated exponential Sheffer are applied on an integral transform to introduce the extended forms of the hybrid Sheffer sequences. The determinant forms and other properties for these sequences are studied via fractional operators and Riordan matrices. 2. Extended Hybrid Sheffer Sequences We show that the combination of exponential operators with the integral transform for the (r) 2VTESS esn (x, y) will give rise to a new class of extended hybrid Sheffer sequences, namely the extended truncated exponential-Sheffer sequences (ETESS). Here, we define the extended hybrid Sheffer sequences by the following definition: Definition 6. The extended hybrid Sheffer sequences are defined by the following operational rule: ¶ ¶r −n a − y y fsn(x)g = (r) sn(x, y; a). (17) ¶y ¶xr ne Theorem 1. For the extended hybrid Sheffer sequences, the following integral representation holds true: Z ¥ 1 −at n−1 (r) e(r) sn(x, y; a) = e t esn (x, yt)dt. (18) n G(n) 0 ! ¶ ¶r Proof. Replacing a by a − y ¶y y ¶xr in integral (2) and then operating the resultant expression on sn(x), we find − ¶ ¶r n 1 Z ¥ ¶ ¶r − ( ) = −at n−1 ( ) a y y r sn x e t exp ty y r sn x dt, (19) ¶y ¶x G(n) 0 ¶y ¶x which in view of Equation (14) gives − ¶ ¶r n 1 Z ¥ − ( ) = −at n−1 (r)( ) a y y r sn x e t esn x, yt dt. (20) ¶y ¶x G(n) 0 Denoting the right hand side of Equation (20) by a new class of extended hybrid Sheffer sequences, i.e., (r) s (x, y; a) yields assertion (18).