Some Identities Associated with 2-Variable Truncated Exponential Based Sheffer Polynomial Sequences

Home , Pascal matrix

Commun. Korean Math. Soc. 35 (2020), No. 2, pp. 533–546 https://doi.org/10.4134/CKMS.c190104 pISSN: 1225-1763 / eISSN: 2234-3024

SOME IDENTITIES ASSOCIATED WITH 2-VARIABLE TRUNCATED EXPONENTIAL BASED SHEFFER POLYNOMIAL SEQUENCES

Junesang Choi, Saima Jabee, and Mohd Shadab

Abstract. Since Sheffer introduced the so-called Sheffer polynomials in 1939, the polynomials have been extensively investigated, applied and classified. In this paper, by using matrix algebra, specifically, some properties of Pascal and Wronskian matrices, we aim to present certain interesting identities involving the 2-variable truncated exponential based Sheffer polynomial sequences. Also, we use the main results to give some interesting identities involving so-called 2-variable truncated exponential based Miller-Lee type polynomials. Further, we remark that a number of different identities involving the above polynomial sequences can be derived by applying the method here to other combined generating functions.

1. Introduction and preliminaries Sequences of polynomials play an important role in dealing with various problems arising in many different areas of pure and applied mathematics (see, e.g., [2,5, 17, 19–22]). Among a variety of polynomials, in 1939, Sheffer [24] introduced the so-called Sheffer polynomials. The Sheffer polynomials have been extensively investigated, applied and classified (see, e.g., [18, pp. 218– 232]). ∞ A polynomial sequence {sn(x)}n=0 is called Sheffer polynomial sequence if and only if its generating function is given by ∞ X tn (1.1) a(t) exp (x b(t)) = s (x) . n n! n=0 Here a(t) and b(t) are formal power series ∞ ∞ X n X n a(t) = an t and b(t) = bn t n=0 n=0

Received March 19, 2019; Accepted October 14, 2019. 2010 Mathematics Subject Classification. 15A15, 15A24, 42C05, 65QXX. Key words and phrases. Sheffer polynomials, truncated exponential-Sheffer polynomial sequences, Pascal matrix, Wronskian matrix, Miller-Lee type polynomials.

c 2020 Korean Mathematical Society 533 534 J. CHOI, S. JABEE, AND M. SHADAB

0 with a(0) = a0 6= 0, b(0) = b0 = 0, and b (0) = b1 6= 0 (see, e.g., [10, 20, 24]). The truncated exponential polynomials en(x) defined by the series (see [3]) n X xk (1.2) e (x) = n k! k=0 are the first n+1 terms of the Maclaurin series for ex. Obviously, the truncated exponential polynomials en(x) are defined by the generating function (see [6]) ∞ ext X (1.3) = e (x) tn. 1 − t n n=0

The higher-order truncated exponential polynomials [r]en(x) deﬁned by the series [ n ] Xr xn−rk e (x) = (n, r ∈ )(1.4) [r] n (n − rk)! N k=0 are generated by the following function ∞ ext X (1.5) = e (x) tn 1 − tr [r] n n=0 (see [6]). Here and in the following, let N and C be the sets of positive integers and complex numbers, respectively, and let N0 := N ∪ {0}. (r) The 2-variable truncated exponential polynomials (2VTEP) en (x, y) of order r deﬁned by

[ n ] Xr ykxn−rk (1.6) e(r)(x, y) = n (n − rk)! k=0 are generated by the following function ∞ ext X tn (1.7) = e(r)(x, y) 1 − ytr n n! n=0 (see [8, p. 174]). Let F be a class of functions which are analytic at the origin. Then the generalized Pascal functional matrix [Pn(g(t))] (g(t) ∈ F) is a lower triangular n + 1 by n + 1 matrix deﬁned by

(i (i−j) j g (t), if i ≥ j (1.8) Pn[g(t)]i,j = 0, otherwise, for all i, j = 0, 1, . . . , n (see [25, 27]). Here and in the following, g(i)(t) is ith derivative of g(t). TRUNCATED EXPONENTIAL BASED SHEFFER POLYNOMIAL SEQUENCE 535

th The n order Wronskian matrix of analytic functions g1(t), g2(t), . . . , gm(t) ∈ F is an n + 1 by m matrix defined by   g1(t) g2(t) g3(t) . . . gm(t)  0 0 0 0   g1(t) g2(t) g3(t) . . . gm(t)    (1.9) Wn[g1(t), g2(t), . . . , gm(t)] =  . . . . .   ......    (n) (n) (n) (n) g1 (t) g2 (t) g3 (t) . . . gm (t) (see [25, 27]). Many authors have presented some recurrence relations, differential equa- tions and identities involving various polynomial sequences (see, e.g., [4,9, 11– 13,16,23]). Youn and Yang [27] obtained some identities and differential equation for Sheffer polynomial sequences by using matrix algebra (see also [1]). Here and in the following, let g(t) be an invertible analytic function, that is, g(0) 6= 0, and f(t) be analytic function with f(0) = 0 and f 0(0) 6= 0 that admits compositional inverse. Then the 2-variable truncated exponential based Sheffer polynomial sequences e(r) sn(x, y) are defined by the following generating function ∞ n 1 −1 X t (1.10) exp(xf (t)) = (r) s (x, y) , g(f −1(t)) (1 − y(f −1(t))r) e n n! n=0 where f −1(t) is the compositional inverse of f(t). In order to define the polynomial sequence in (1.10), the left-member of (1.10) should be analytic at t = 0. Then, obviously, n −1 d exp (xf (t)) (1.11) e(r) sn(x, y) = n −1 −1 r (n ∈ N0) . dt g(f (t))(1 − y(f (t)) ) t=0 Khan et al. [14] used operational methods to present some useful identities involving the 2-variable truncated exponential based Sheffer polynomial sequences e(r) sn(x, y) and apply some results to demonstrate some special polynomials, for example, the 2-variable truncated-exponential based generalized Laguerre polynomials. Youn and Yang [27] used matrix algebra to provide a differential equation and several recursive formulas of one variable Sheffer polynomial sequences. In this paper, by employing Youn and Yang’s method [27], we derive some presumably new identities involving the 2-variable truncated exponential based Sheffer polynomial sequences e(r) sn(x, y). To do this, some properties of the Wronskian matrices and the generalized Pascal functional matrices and their relationships are recalled in the following lemma (see, e.g., [26, 27]).

Lemma 1. Let u, v ∈ C and g(t), g1(t), . . . , gm(t), h(t) ∈ F. Then (a) Linear

Pn[u g(t) + v h(t)] = u Pn[g(t)] + v Pn[h(t)],

Wn[u g(t) + v h(t)] = u Wn[g(t)] + v Wn[h(t)]. 536 J. CHOI, S. JABEE, AND M. SHADAB

(b) Multiplicative

Pn[g(t) h(t)] = Pn[g(t)] Pn[h(t)] = Pn[h(t)] Pn[g(t)]. −1 −1 −1 In addition, if g(t) 6= 0, then (Pn[g(t)]) = Pn g (t) , where g (t) denotes the multiplicative inverse of g(t). (c) Pascal and Wronskian

Pn[g(t)] Wn[h(t)] = Pn[h(t)] Wn[g(t)] = Wn[(gh)(t)]. In addition,

Pn[g(t)] Wn[g1(t), g2(t), . . . , gm(t)] = Wn[(g g1)(t), (g g2)(t),..., (g gm)(t)]. (d) Let g(0) = 0 and g0(0) 6= 0. Then 2 3 n −1 Wn[h(g(t))]t=0 = Wn 1, g(t), g (t), g (t), . . . , g (t) t=0 Ωn Wn[h(t)]t=0.

Here and in the following, Ωn := diag[0!, 1!, 2!, . . . , n!] is the diagonal n + 1 by n + 1 matrix.

2. Some identities involving 2-variable truncated exponential based Sheffer polynomial sequences Here, we introduce a vector form of the 2-variable truncated exponential based Sheffer polynomial sequences e(r) sn(x, y) for (g(t), f(t)) which is defined by T (2.1) e(r~) sn(x, y) := [e(r) s0(x, y), e(r) s1(x, y),..., e(r) sn(x, y)] , where T denotes the transpose of a matrix. From (1.11), we have 1 −1 (2.2) e(r~) sn(x, y) = Wn −1 −1 r exp(xf (t)) . g(f (t))(1 − y(f (t)) ) t=0

Lemma 2. Let e(r) sn(x, y) be the 2-variable truncated exponential based Sheﬀer polynomial sequences for (g(t), f(t)). Then T −1 Wn[e(r) s0(x, y), e(r) s1(x, y),..., e(r) sn(x, y)] Ωn = W 1, f −1(t), (f −1(t))2,..., (f −1(t))n Ω−1 (2.3) n t=0 n 1 xt × Pn r Pn e t=0 . g(t)(1 − yt ) t=0 Proof. Applying (d) in Lemma1 to the right member of (2.2), we obtain −1 −1 2 −1 n −1 e(r~) sn(x, y) = Wn 1, f (t), (f (t)) ,..., (f (t)) t=0 Ωn (2.4) ext × Wn r . g(t)(1 − yt ) t=0 Using (c) in Lemma1, we get xt e 1 xt (2.5) Wn r = Pn r Wn e t=0. g(t)(1 − yt ) t=0 g(t)(1 − yt ) t=0 TRUNCATED EXPONENTIAL BASED SHEFFER POLYNOMIAL SEQUENCE 537

Easily, we ﬁnd the following well known identity xt 2 n T (2.6) Wn e t=0 = [1 x x ··· x ] . Using (2.6) in (2.5) and applying the resulting identity in (2.4), we have −1 −1 2 −1 n −1 e(r~) sn(x, y) = Wn 1, f (t), (f (t)) ,..., (f (t)) t=0 Ωn (2.7) 1 2 n T × Pn r [1 x x ··· x ] . g(t)(1 − yt ) t=0 Taking kth order partial derivative with respect to x on both sides of (2.7) and dividing each side of the resulting identity by k!, we obtain (2.8) T 1 ∂k ∂k ∂k (r) s (x, y), (r) s (x, y),..., (r) s (x, y) k! ∂xk e 0 ∂xk e 1 ∂xk e n −1 −1 2 −1 n −1 1 = Wn 1, (f (t)), (f (t)) ,..., (f (t)) t=0 Ωn Pn r g(t)(1 − yt ) t=0 k + 1 k + 2 n T × 0 ··· 0 1 x x2 ··· xn−k . k k k Finally, we observe that the right member and the left member of (2.8) are equal to the kth columns of the corresponding member of (2.3), respectively. This completes the proof.

Theorem 3. Let e(r) sn(x, y) be the 2-variable truncated exponential based Shef- fer polynomial sequences for (g(t), f(t)). Then

n k X (xαk + yrβk + γk) ∂ (2.9) (r) s (x, y) = (r) s (x, y), k! ∂xk e n e n+1 k=0 where (k) 1 αk = 0 (k ∈ N0), f (t) t=0 r−1 (k) t βk = r 0 (k ∈ N0) (1 − yt )f (t) t=0 and 0 (k) g (t) γk = − 0 (k ∈ N0). g(t)f (t) t=0 Proof. Consider −1 d exp (xf (t)) (2.10) A(t; x, y; n, r) t=0 := Wn −1 −1 r . dt g(f (t))(1 − y(f (t)) ) t=0 Using (1.9), we get T (2.11) A(t; x, y; n, r) t=0 = [e(r) s1(x, y), e(r) s2(x, y),..., e(r) sn+1(x, y)] . 538 J. CHOI, S. JABEE, AND M. SHADAB

Since f 0(0) 6= 0, we have

−1 0 1 (2.12) f (t) = 0 −1 . t=0 f (f (t)) t=0 Using (2.12), we ﬁnd (2.13)

A(t; x, y; n, r) t=0 " ! 1 (f −1(t))r−1 g0(f −1(t)) = W x + yr − n f 0(f −1(t)) (1 − y(f −1(t))r)f 0(f −1(t)) g(f −1(t))f 0(f −1(t)) # exp (xf −1(t)) × . g(f −1(t))(1 − y(f −1(t))r) t=0 Using (d) in Lemma1, we obtain

A(t; x, y; n, r) t=0 = W 1, f −1(t), (f −1(t))2,..., (f −1(t))n Ω−1 (2.14) n t=0 n 1 tr−1 g0(t) exp (xt) × Wn x 0 + yr r 0 − 0 r . f (t) (1 − yt )f (t) g(t)f (t) g(t)(1 − yt ) t=0 Employing (b) and (c) in Lemma1, we get (2.15) −1 −1 2 −1 n −1 A(t; x, y; n, r) t=0 = Wn 1, (f (t)), (f (t)) ,..., (f (t)) t=0 Ωn 1 × Pn r Pn [exp (xt)]t=0 g(t)(1 − yt ) t=0 1 tr−1 g0(t) × Wn x 0 + yr r 0 − 0 . f (t) (1 − yt )f (t) g(t)f (t) t=0 Using Lemma2, we obtain T −1 A(t; x, y; n, r) t=0 = Wn[e(r) s0(x, y), e(r) s1(x, y),..., e(r) sn(x, y)] Ωn 1 tr−1 g0(t) × Wn x 0 + yr r 0 − 0 . f (t) (1 − yt )f (t) g(t)f (t) t=0 Or, equivalently, (2.16)

A(t; x, y; n, r) t=0   e(r) s0(x, y) 0 0 ... 0 0  s (x,y)    e(r) 1 xα0 + yrβ0 + γ0  (r) s (x, y) 0 ... 0  e 1 1!  xα + yrβ + γ 0 00  1 1 1   s (x,y) s (x,y)     e(r) 2 e(r) 2  xα2 + yrβ2 + γ2 = e(r) s2(x, y) ... 0   .  1! 2!   .   . . . . .   .   ......   .   . . . .  0 00 xαn + yrβn + γn  s (x,y) s (x,y) s (n)(x,y)  e(r) n e(r) n e(r) n e(r) sn(x, y) 1! 2! ... n! TRUNCATED EXPONENTIAL BASED SHEFFER POLYNOMIAL SEQUENCE 539

Finally, equating nth rows of the right members of (2.11) and (2.16), we obtain the desired result (2.9).

Theorem 4. Let e(r) sn(x, y) be the 2-variable truncated exponential based Shef- fer polynomial sequences for (g(t), f(t)). Then n X n (2.17) (r) s (x, y) = (xδ + yr + ζ ) (r) s (x, y), e n+1 k n−k n−k n−k e k k=0 where (k) 1 δk = 0 −1 (k ∈ N0), f (f (t)) t=0

−1 r−1 (k) (f (t)) k = −1 r 0 −1 (k ∈ N0) (1 − y(f (t)) )f (f (t)) t=0 and 0 −1 (k) g (f (t)) ζk = − −1 0 −1 (k ∈ N0). g(f (t))f (f (t)) t=0 Proof. Applying (c) in Lemma1 to (2.13), we obtain

A(t; x, y; n, r) t=0 1 (f −1(t))r−1 g0(f −1(t)) = Pn x 0 −1 + yr −1 r 0 −1 − −1 0 −1 f (f (t)) (1 − y(f (t)) )f (f (t)) g(f (t))f (f (t)) t=0 exp (f −1(t)) × Wn −1 −1 r . g(f (t))(1 − y(f (t)) ) t=0 Using (a) in Lemma1 together with (2.1) and (2.2), we have (2.18)

A(t; x, y; n, r) t=0   xδ0 + yr0 + ζ0 0 0 ... 0  xδ + yr + ζ xδ + yr + ζ 0 ... 0   1 1 1 0 0 0   2   xδ2 + yr2 + ζ2 (xδ1 + yr1 + ζ1) xδ0 + yr0 + ζ0 ... 0  =  1   . . . . .   ......   . . . .  n xδn + yrn + ζn 1 (xδn−1 + yrn−1 + ζn−1) ...... xδ0 + yr0 + ζ0   e(r) s0(x, y)  e(r) s1(x, y)     .  ×  .  .   e(r) sn−1(x, y) e(r) sn(x, y) Finally, equating nth rows of right members of (2.11) and (2.18), we obtain the desired result. 540 J. CHOI, S. JABEE, AND M. SHADAB

Theorem 5. Let e(r) sn(x, y) be the 2-variable truncated exponential based Shef- fer polynomial sequences for (g(t), f(t)). Then n X e(r) sn−k+1(x, y)ηk k=0 (2.19) n X n = x (r) s (x, y) + (yrθ + ϑ ) (r) s (x, y), e n k k k e n−k k=0 where

0 −1 (k) ηk = f (f (t)) (k ∈ N0), t=0 −1 r−1 (k) (f (t)) θk = −1 r (k ∈ N0) (1 − y(f (t)) ) t=0 and 0 −1 (k) g (f (t)) ϑk = − −1 (k ∈ N0). g(f (t)) t=0 Proof. Consider

B(t; x, y; n, r) t=0 (2.20) −1 0 −1 d exp (xf (t)) := Wn f (f (t)) −1 −1 r . dt g(f (t))(1 − y(f (t)) ) t=0 Using (c) in Lemma1, we get

B(t; x, y; n, r) t=0 −1 d exp (xf (t)) 0 −1 = Pn −1 −1 r Wn f (f (t)) t=0 . dt g(f (t))(1 − y(f (t)) ) t=0 We ﬁnd from (2.11) that (2.21)

B(t; x, y; n, r) t=0   e(r) s1(x, y) 0 0 ... 0   η0  (r) s (x, y) (r) s (x, y) 0 ... 0   e 2 e 1  η1   2     e(r) s3(x, y) e(r) s2(x, y) e(r) s1(x, y)) ... 0  η2  =  1    .  . . . . .   .   ......   .   . . . .  n n ηn e(r) sn+1(x, y) 1 e(r) sn(x, y) 2 e(r) sn(x, y) ... e(r) s1(x, y) We also have (2.22)

B(t; x, y; n, r) t=0 (f −1(t))r−1 g0(f −1(t)) exp (xf −1(t)) = Wn x + yr −1 r − −1 −1 −1 r , (1 − y(f (t)) ) g(f (t)) g(f (t))(1 − y(f (t)) ) t=0 TRUNCATED EXPONENTIAL BASED SHEFFER POLYNOMIAL SEQUENCE 541 which, upon using (a), (b) and (c) in Lemma1, yields

B(t; x, y; n, r) t=0 exp (xf −1(t)) = xWn −1 −1 r g(f (t))(1 − y(f (t)) ) t=0 (2.23) exp (xf −1(t)) (f −1(t))r−1 + yrPn −1 −1 r Wn −1 r g(f (t))(1 − y(f (t)) ) t=0 (1 − y(f (t)) ) t=0 exp (xf −1(t)) g0(f −1(t)) + Pn −1 −1 r Wn − −1 . g(f (t))(1 − y(f (t)) ) t=0 g(f (t)) t=0

Finally, equating nth rows of the right members of (2.21) and (2.23), we obtain the desired result (2.19).

Theorem 6. Let e(r) sn(x, y) be the 2-variable truncated exponential based Shef- fer polynomial sequences for (g(t), f(t)). Then

n k X (xκk + yrλk + µk) ∂ (2.24) n (r) s (x, y) = (r) s (x, y), e n k! ∂xk e n−k k=0 where (k) f(t) κk = 0 (k ∈ N0), f (t) t=0

r−1 (k) t f(t) λk = r 0 (k ∈ N0) (1 − yt )f (t) t=0 and

0 (k) g (t)f(t) µk = − 0 (k ∈ N0). g(t)f (t) t=0 Proof. Consider

−1 d exp (xf (t)) (2.25) C(t; x, y; n, r) t=0 := Wn t −1 −1 r . dt g(f (t))(1 − y(f (t)) ) t=0

Using (c) in Lemma1, we have

−1 d exp (xf (t)) C(t; x, y; n, r) t=0 = Pn [t] Wn −1 −1 r . t=0 dt g(f (t))(1 − y(f (t)) ) t=0 542 J. CHOI, S. JABEE, AND M. SHADAB

Or, equivalently,

C(t; x, y; n, r) t=0 0 0 0 ... 0 0 0   2 0 0 ... 0 0 0 e(r) s1(x, y)      e(r) s2(x, y)  0 3 0 ... 0 0 0   (2.26)    e(r) s3(x, y)  = 0 0 4 ... 0 0 0   .    .  ......   .  ......    ......   e(r) sn(x, y)    0 0 0 . . . n 0 0 e(r) sn+1(x, y) 0 0 0 ... 0 n + 1 0 We also have

C(t; x, y; n, r) t=0 " f(f −1(t)) (f −1(t))r−1f(f −1(t)) g0(f −1(t))f(f −1(t)) = W x + yr − n f 0(f −1(t)) (1 − y(f −1(t))r)f 0(f −1(t)) g(f −1(t))f 0(f −1(t)) # exp (xf −1(t)) × , g(f −1(t))(1 − y(f −1(t))r) t=0 which, upon using (d) in Lemma1, yields

C(t; x, y; n, r) t=0 = W 1, f −1(t), (f −1(t))2,..., (f −1(t))n Ω−1 (2.27) n t=0 n f(t) tr−1f(t) g0(t)f(t) exp (xt) × Wn x 0 + yr r 0 − 0 r . f (t) (1 − yt )f (t) g(t)f (t) g(t)(1 − yt ) t=0 Applying (b) and (c) in (2.27), we get

C(t; x, y; n, r) t=0 −1 −1 2 −1 n −1 1 = Wn 1, f (t), (f (t)) ,..., (f (t)) Ωn Pn r t=0 g(t)(1 − yt ) t=0 r−1 0 f(t) t f(t) g (t)f(t) × Pn [exp (xt)]t=0 Wn x 0 + yr r 0 − 0 , f (t) (1 − yt )f (t) g(t)f (t) t=0 which, in view of Lemma2, leads to

C(t; x, y; n, r) t=0 T −1 = W [ (r) s (x, y), (r) s (x, y), (r) s (x, y),..., (r) s (x, y)] Ω (2.28) n e 0 e 1 e 2 e n n r−1 0 f(t) t f(t) g (t)f(t) × Wn x 0 + yr r 0 − 0 . f (t) (1 − yt )f (t) g(t)f (t) t=0 Finally, equating nth rows of the right members of (2.26) and (2.28), we obtain the desired result (2.24). TRUNCATED EXPONENTIAL BASED SHEFFER POLYNOMIAL SEQUENCE 543

3. An application (m) Andrews [3, p. 320] commented that the Miller-Lee polynomials Gn (x) given by

n X m + n − k xk (3.1) G(m)(x) = n m k! k=0 arise in the problem of ﬁnding the probability density function for the output of a cross correlator, by referring to two related papers. (m) Miller-Lee polynomials Gn (x) are given by the following generating function (see [7, p. 21, Eq. (1.11)] and [15, p. 760, Eq. (2.14)])

∞ 1 X (3.2) exp(xt) = G(m)(x) tn (|t| < 1). (1 − t)m+1 n n=0 Dattoli et al. [7, p. 21, Eq. (1.9)], among integral representations of some other polynomials, presented the following integral representation Z ∞ (m) 1 −t m n (3.3) Gn (x) = e t (x + t) dt. m! n! 0 Khan et al. [15, p. 760, Eq. (2.15)]) introduced Hermite-Miller-Lee polyno- (m) mials H Gn (x, y, z) deﬁned by the following generating function

∞ 1 X (3.4) exp xt + yt2 + zt3 = G(m)(x, y, z) tn (|t| < 1). (1 − t)m+1 H n n=0 They also [15, p. 760, Eqs. (2.16) and (2.17)]) pointed out that the cases m = 0 and m = α − 1 of (3.4) give the following generating functions for, respec- (m) tively, Hermite-truncated exponential polynomials H en (x, y, z) and Hermite- (α) modiﬁed Laguerre polynomials H fn (x, y, z):

∞ 1 X (3.5) exp xt + yt2 + zt3 = e(m)(x, y, z) tn (|t| < 1) 1 − t H n n=0 and ∞ 1 X (3.6) exp xt + yt2 + zt3 = f (α)(x, y, z) tn (|t| < 1). (1 − t)α H n n=0

From Section2, let e(r) sn(x, y) be the 2-variable truncated exponential based 1 Sheﬀer polynomial sequences for (1−t)m+1 , t , which we call 2-variable trun- (m) cated exponential based Miller-Lee type polynomials e(r) Gn (x, y). 544 J. CHOI, S. JABEE, AND M. SHADAB

(m) Applying the results in Theorems3-6 to e(r) Gn (x, y), we, respectively, get the following identities: (3.7) (m) 1 r (m) e(r) Gn+1(x, y) = x + ry − m − 1 e(r) Gn (x, y) n k X 1+k ∂ (m) + ry r − m − 1 (r) G (x, y)(n ∈ ); ∂xk e n N0 k=1 (3.8) (m) e(r) Gn+1(x, y) 1 r (m) = ry − m − 1 e(r) Gn (x, y) n−1 X n 1+n−k (m) + x + r(n − k)!y r −(m+1)(n−k)! (r) G (x, y)(n ∈ ); k e n−k N0 k=0

n (m) (m) X (m) e(r) Gn+1(x, y) − xe(r) Gn (x, y) + e(r) Gn−k+1(x, y) (3.9) k=1 n X n 1+k (m) = ry r − (m + 1) k! (r) G (x, y)(n ∈ ); k e n−k N0 k=0

(m) (n − r)e(r) Gn (x, y)

1 ∂ (m) = x + ry r − m − 1 (r) G (x, y) (3.10) ∂x e n−1 n k X 1 k ∂ (m) + x + ry r k! − (m + 1)k! (r) G (x, y)(n ∈ ) . k! ∂xk e n−k N0 k=2 4. Remarks In addition to the results in Theorems3-6, we can obtain a number of identities involving the 2-variable truncated exponential based Sheﬀer polynomial sequences for (g(t), f(t)) by considering some combinations diﬀerent from

A(t; x, y; n, r) t=0, B(t; x, y; n, r) t=0, and C(t; x, y; n, r) t=0, for example, −1 0 −1 −1 d exp (xf (t)) Wn f (f (t))g(f (t)) −1 −1 r , dt g(f (t))(1 − y(f (t)) ) t=0 1 Wn −1 −1 r g(f (t))(1 − y(f (t)) ) t=0 −1 exp (xf (t)) −1 = Wn −1 −1 r exp (−xf (t)) , g(f (t))(1 − y(f (t)) ) t=0 −1 −1 d exp (xf (t)) Wn g(f (t)) −1 −1 r , dt g(f (t))(1 − y(f (t)) ) t=0 TRUNCATED EXPONENTIAL BASED SHEFFER POLYNOMIAL SEQUENCE 545

exp (xf −1(t)) Wn −1 r (1 − y(f (t)) ) t=0 −1 −1 exp (xf (t)) = Wn g(f (t)) −1 −1 r , g(f (t))(1 − y(f (t)) ) t=0 and −1 0 −1 d exp (xf (t)) Wn tf (f (t)) −1 −1 r . dt g(f (t))(1 − y(f (t)) ) t=0

References [1] L. Aceto and I. Ca¸cão, A matrix approach to Sheffer polynomials, J. Math. Anal. Appl. 446 (2017), no. 1, 87–100. https://doi.org/10.1016/j.jmaa.2016.08.038 [2] G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999. https://doi. org/10.1017/CBO9781107325937 [3] L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Co., New York, 1985. [4] S. Araci, Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus, Appl. Math. Comput. 233 (2014), 599–607. https: //doi.org/10.1016/j.amc.2014.01.013 [5] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach Science Publishers, New York, 1978. [6] G. Dattoli, C. Cesarano, and D. Sacchetti, A note on truncated polynomials, Appl. Math. Comput. 134 (2003), no. 2-3, 595–605. https://doi.org/10.1016/S0096-3003(01) 00310-1 [7] G. Dattoli, S. Lorenzutta, and D. Sacchetti, Integral representations of new families of polynomials, Ital. J. Pure Appl. Math. No. 15 (2004), 19–28. [8] G. Dattoli, M. Migliorati, and H. M. Srivastava, A class of Bessel summation formulas and associated operational methods, Fract. Calc. Appl. Anal. 7 (2004), no. 2, 169–176. [9] , Sheffer polynomials, monomiality principle, algebraic methods and the theory of classical polynomials, Math. Comput. Modelling 45 (2007), no. 9-10, 1033–1041. https://doi.org/10.1016/j.mcm.2006.08.010 [10] A. Di Bucchianico and D. Loeb, A selected survey of umbral calculus, Electron. J. Combin. 2 (1995), Dynamic Survey 3, 28 pp. [11] L. G. Garza, L. E. Garza, F. Marcellán,and N. C. Pinzón-Cortés, A matrix approach for the semiclassical and coherent orthogonal polynomials, Appl. Math. Comput. 256 (2015), 459–471. https://doi.org/10.1016/j.amc.2015.01.071 [12] Y. He, S. Araci, H. M. Srivastava, and M. Acikgoz, Some new identities for the Apostol- Bernoulli polynomials and the Apostol-Genocchi polynomials, Appl. Math. Comput. 262 (2015), 31–41. https://doi.org/10.1016/j.amc.2015.03.132 [13] M. X. He and P. E. Ricci, Differential equation of Appell polynomials via the factor- ization method, J. Comput. Appl. Math. 139 (2002), no. 2, 231–237. https://doi.org/ 10.1016/S0377-0427(01)00423-X [14] S. Khan, G. Yasmin, and N. Ahmad, On a new family related to truncated exponential and Sheffer polynomials, J. Math. Anal. Appl. 418 (2014), no. 2, 921–937. https: //doi.org/10.1016/j.jmaa.2014.04.028 [15] S. Khan, G. Yasmin, R. Khan, and N. A. M. Hassan, Hermite-based Appell polynomials: properties and applications, J. Math. Anal. Appl. 351 (2009), no. 2, 756–764. https: //doi.org/10.1016/j.jmaa.2008.11.002 546 J. CHOI, S. JABEE, AND M. SHADAB

[16] D. S. Kim and T. Kim, A matrix approach to some identities involving Sheffer polynomial sequences, Appl. Math. Comput. 253 (2015), 83–101. https://doi.org/10.1016/ j.amc.2014.12.048 [17] D. H. Lehmer, A new approach to Bernoulli polynomials, Amer. Math. Monthly 95 (1988), no. 10, 905–911. https://doi.org/10.2307/2322383 [18] E. D. Rainville, Special Functions, The Macmillan Co., New York, 1960. [19] S. Roman, The theory of the umbral calculus. I, J. Math. Anal. Appl. 87 (1982), no. 1, 58–115. https://doi.org/10.1016/0022-247X(82)90154-8 [20] , The Umbral Calculus, Pure and Applied Mathematics, 111, Academic Press, Inc., New York, 1984. [21] S. M. Roman and G.-C. Rota, The umbral calculus, Advances in Math. 27 (1978), no. 2, 95–188. https://doi.org/10.1016/0001-8708(78)90087-7 [22] K. Rosen, Handbook of Discrete and Combinatorial Mathematics, CRC Press, Boca Raton, Florida, 2000. [23] I. M. Sheffer, A differential equation for Appell polynomials, Bull. Amer. Math. Soc. 41 (1935), no. 12, 914–923. https://doi.org/10.1090/S0002-9904-1935-06218-4 [24] , Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590– 622. http://projecteuclid.org/euclid.dmj/1077491414 [25] Y. Yang and C. Micek, Generalized Pascal functional matrix and its applications, Linear Algebra Appl. 423 (2007), no. 2-3, 230–245. https://doi.org/10.1016/j.laa.2006.12. 014 [26] Y. Yang and H. Youn, Appell polynomial sequences: a linear algebra approach, JP J. Algebra Number Theory Appl. 13 (2009), no. 1, 65–98. [27] H.-Y. Youn and Y.-Z. Yang, Differential equation and recursive formulas of Sheffer polynomial sequences, Discrete Math. 2011 (2011), Article ID 476462, 1–16.

Junesang Choi Department of Mathematics Dongguk University Gyeongju 38066, Korea 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]

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]