Finding Determinant Forms of Certain Hybrid Sheffer Sequences

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 Γ-function [4] is given by

∞ Z Γ(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−ν = e−attν−1dt, min{Re(ν), Re(a)} > 0. (2) Γ(ν) 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 ∈ K for all k ∈ N := {0, 1, 2, ...}. 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))n∈N 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 ∈ K, where n being the degree of polynomial and K a ﬁeld of characteristic zero.

Let (sn(x))n∈N 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)):

√ ∞ 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 {xn} 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 {sn(x)}. (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 {An(x)}. (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 −ν α − y y {sn(x)} = (r) sn(x, y; α). (17) ∂y ∂xr νe

Theorem 1. For the extended hybrid Sheffer sequences, the following integral representation holds true:

Z ∞ 1 −αt ν−1 (r) e(r) sn(x, y; α) = e t esn (x, yt)dt. (18) ν Γ(ν) 0 ! ∂ ∂r Proof. Replacing a by α − y ∂y y ∂xr in integral (2) and then operating the resultant expression on sn(x), we ﬁnd

− ∂ ∂r ν 1 Z ∞ ∂ ∂r − ( ) = −αt ν−1 ( ) α y y r sn x e t exp ty y r sn x dt, (19) ∂y ∂x Γ(ν) 0 ∂y ∂x which in view of Equation (14) gives

− ∂ ∂r ν 1 Z ∞ − ( ) = −αt ν−1 (r)( ) α y y r sn x e t esn x, yt dt. (20) ∂y ∂x Γ(ν) 0

Denoting the right hand side of Equation (20) by a new class of extended hybrid Sheffer sequences, i.e., (r) s (x, y; α) yields assertion (18). νe n

Theorem 2. For the extended hybrid Sheffer sequences (r) s (x, y; α), the following generating function νe n holds true: ex f (w) ∞ wn = (r) sn(x, y; α) . (21) r ν ∑ νe g( f (w))(α − yDyy( f (w)) ) n=0 n! Mathematics 2019, 7, 1105 5 of 16

wn Proof. Multiplying both sides of Equation (18) by n! and summing over n, we ﬁnd

∞ n ∞ Z ∞ n w 1 −αt ν−1 (r) w (r) sn(x, y; α) = e t esn (x, yt) dt, (22) ∑ νe ∑ ( ) n=0 n! n=0 Γ ν 0 n! which on using Equation (13) in the right hand side gives

∞ n x f (w) Z ∞ r w e − α−yDyy( f (w)) t ν−1 (r) s (x, y; α) = e t dt. (23) ∑ νe n n=0 n! Γ(ν)g( f (w)) 0

Making use of Equation (2) in the right hand side of the above equation assertion (21) is obtained.

A recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given; each further term of the sequence or array is defined as a function of the preceding terms. Differentiating generating function (21), with respect to x, y and α, we find the following differential recurrence relations for the extended hybrid Sheffer sequences:

f (D ) { (r) s (x, y; α)} = n (r) s − (x, y; α), (24) x νe n νe n 1 f Dy { (r) sn(x, y; α)} = ν n(n − 1) ... (n − r + 1) (r) sn−r(x, y; α), (25) νe ν+1e

Dα{ (r) sn(x, y; α)} = −ν (r) sn(x, y; α), (26) νe ν+1e The combination of monomiality principle [21] with operational methods opens new possibilities to deal with the theoretical foundations of special polynomials and also to introduce new families of special polynomials. The Sheffer sequences are quasi-monomial. To frame the extended hybrid Sheffer sequences within the context of monomiality principle, the following result is proved:

Theorem 3. The extended hybrid Sheffer sequences (r) s (x, y; α) are quasi-monomial with respect to the νe n following multiplicative and derivative operators:

∂r g0(D ) 1 Mˆ = x − ryD y − x (r) s y r−1 0 (27) νe ∂α∂x g(Dx) f (Dx) and ˆ = ( ) P (r) s f Dx , (28) νe respectively.

Proof. We recall the following recurrence relations for the truncated-Sheffer polynomials e(r) sn+1(x, y) from [19]: 0 r−1 g (Dx) 1 x + ryDyyDx − 0 e(r) sn(x, y) = e(r) sn+1(x, y) (29) g(Dx) f (Dx) and

f (Dx){e(r) sn(x, y)} = n e(r) sn−1(x, y), n ≥ 1. (30) ( ) 1 −at ν−1 Consider the operation: Θ : Replacement of y by yt, multiplication by Γ(ν) e t and then integration with respect to t from t = 0 to t = ∞. Now, operating (Θ) on Equation (29) and then using relation (18) with (26) and further in view of recurrence relation Mˆ {pn(x)} = pn+1(x) assertion (27) follows. Again, operating (Θ) on Equation (30) and then using relation (18) and further in view of recurrence relation Pˆ{pn(x)} = npn−1(x) assertion (28) follows. Mathematics 2019, 7, 1105 6 of 16

Remark 2. Using expressions (27) and (28) of the operators in monomiality principle equation Mˆ Pˆ{pn(x)} = n pn(x), we deduce the following consequence of Theorem 3:

Corollary 1. The extended hybrid Sheffer sequences (r) s (x, y; α) satisfy the following differential equation: νe n

∂r g0(D ) f (D ) − − x x − ( ) = x ryDyy r−1 0 n e(r) sn x, y; α 0. (31) ∂α∂x g(Dx) f (Dx) ν

Remark 3. Taking

0 g (Dx) n f (w) = f (w) = w, g(w) = 1, = 0 and sn(x) = x (32) g(Dx) (r) so that (r) s (x, y; α) = e (x, y; α), νe n ν n in extended hybrid Sheffer sequences (r) s (x, y; α), we obtain as a special case the extended truncated νe n (r) exponential polynomials νen (x, y; α). The corresponding results are given in Table1.

(r) Table 1. Results for νen (x, y; α).

S. No. Results Expressions −ν ∂ ∂r n (r) I. Operational rule α − y ∂y y ∂xr x = νen (x, y; α) ∞ exw = (r)( ) wn II. Generating function (α−yD y wr )ν ∑ νen x, y; α n! y n=0 ˆ ∂r ˆ III. Multiplicative and M (r) = x − ryDyy r−1 , P (r) = Dx νen ∂α∂x ν en derivative operators

∂r+1 (r) IV. Differential equation xDx − ryDyy ∂α∂xr − n νen (x, y; α) = 0 −1 Note. It should be noted that for α = ν = 1 and y = Dy , the extended truncated-exponential-Sheffer (r) polynomials (r) sn(x, y; α) reduce to 2VTESP es (x, y). For the same choice of parameters α, ν and variable y νe n (r) (r) the extended truncated-exponential polynomials νen (x, y; α) reduce to the 2VTEP en (x, y).

To establish the determinant form for the extended hybrid Sheffer sequences, the following result is proved:

Theorem 4. For the extended hybrid Sheffer sequences (r) s (x, y; α) of degree n, the following holds: νe n

1 (r) s (x, y; α) = , νe 0 b0,0 ( ) ( ) ( ) ( ) e r (x y ) e r (x y ) ··· e r (x y ) e r (x y ) 1 ν 1 , ; α ν 2 , ; α ν n−1 , ; α ν n , ; α

b b b ··· b − b 0,0 1,0 2,0 n 1,0 n,0

n (−1) 0 b1,1 b2,1 ··· bn−1,1 bn,1 (33) (r) sn(x, y; α) = , νe 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) . Mathematics 2019, 7, 1105 7 of 16

−ν ∂ ∂r Proof. Operating O : exp α − y ∂y y ∂xr on both sides of Equation (6) and then using operational rule (17) and operational rule given in Table1(I), for n = 0, 1, 2, ... in the LHS and RHS, respectively of the resulting equation assertion (33) follows.

Remark 4. Taking

f (w) = f (w) = w and g( f (w)) = g(w); (r) s (x, y; α) = (r) A (x, y; α), νe n νe n in extended hybrid Sheffer sequences (r) s (x, y; α), we get the extended hybrid Appell sequences (r) A (x, y; α). νe n νe n The corresponding results are given in Table2.

Table 2. Results for (r) A (x, y; α). νe n

S. No. Results Expressions −ν ∂ ∂r I. Operational rule α − y y r {A (x)} = (r) A (x, y; α) ∂y ∂x n νe n ∞ exw wn II. Generating function = (r) A (x, y; α) g(w)(α−yD y wr )ν ∑ νe n n! y n=0 r 0 ˆ = − ∂ − g (Dx ) ˆ = III. Multiplicative and M (r) A x ryDyy r−1 , P (r) A Dx νe ∂α∂x g(Dx ) νe derivative operators

r+1 0 ∂ g (Dx ) IV. Differential equation xDx − ryDyy r − Dx − n (r) An(x, y; α) = 0 ∂α∂x g(Dx ) νe

For the extended hybrid Appell sequences (r) A (x, y; α) of degree n, the following holds: νe n

1 (r) A (x, y; α) = , νe 0 g0 (r) (r) (r) (r) 1 e (x, y; α) e (x, y; α) ··· e (x, y; α) e (x, y; α) ν 1 ν 2 ν n−1 ν n

g g g ··· g − gn 0 1 2 n 1

n 2 n−1 n (34) (−1) 0 g0 (1) g1 ··· ( 1 ) gn−2 (1) gn−1 (r) A (x, y; α) = + . ν e n g n 1 0

0 0 g ··· (n−1) g (n) g 0 2 n−3 2 n−2

... ··· ..

... ··· .. n 0 0 0 ··· g0 (n−1) g1

0 g (Dx) Remark 5. Taking g( f (w)) = 1, = 0; an = δn and sn(x) = pn(x) in the results for extended g(Dx) ,0 ,0 hybrid Sheffer sequences (r) s (x, y; α), we get the corresponding results for the extended hybrid associated νe n Sheffer sequences (r) p (x, y; α). νe n

In the next section, examples of some members belonging to the extended hybrid Sheffer and Appell families are considered. Mathematics 2019, 7, 1105 8 of 16

3. Examples By making suitable selections for the pair of function (g(t), f (t)) in the results derived for extended hybrid Sheffer sequences, we obtain results for the particulars members of the extended hybrid Sheffer family. The following examples illustrate this process:

Example 1. Taking

1 w g(w) = , f (w) = f¯(w) = and g( f¯(w)) = (1 − w)β+1 (1 − w)β+1 w − 1 in extended hybrid Sheffer sequences (r) s (x, y; α), we get the extended hybrid associated Laguerre sequences νe n (β) (EhLS) (r) L (x, y; α). The corresponding results are given in Table3. νe n

(β) Table 3. Results for (r) L (x, y; α). νe n

S. No. Results Expressions −ν ∂ ∂r (β) (β) I. Operational rule α − y y r {L (x)} = (r) L (x, y; α) ∂y ∂x n νe n

w ∞ n exp(x( w−1 )) (β) w II. Generating function + w = (r) L (x, y; α) (1−w)β 1(α−yD y( )r )ν ∑ νe n n! y w−1 n=0 r ( + ) ˆ = − ∂ − β 1 −1 ˆ = Dx III. Multiplicative and M (r) L x ryDyy r−1 −2 , P (r) L − νe ∂α∂x (1−Dx ) (1−Dx ) νe Dx 1 derivative operators

r ∂ (β+1) −Dx (β) IV. Differential equation x − ryDyy r−1 − −1 − n (r) Ln (x, y; α) = 0 ∂α∂x (1−Dx ) (1−Dx ) νe

(β) For the extended hybrid associated Laguerre sequences (r) L (x, y; α) of degree n, the following holds: νe n

(β) (r) L (x, y; α) = 1, νe 0

(r) (r) (r) (r) 1 e (x, y; α) e (x, y; α) ··· e (x, y; α) e (x, y; α) ν 1 ν 2 ν n−1 ν n

1 (β + 1)(β + 2)2 ··· (β + n − 1)n−1 (β + n)n

0 −1 −2(β + 2) · · · −(n − 1)(β + n − 1)n−2 −n(β + n)n−1 (35) (β) n(n+3) (r) L (x, y; α) = (−1) 2 . νe n (n− )(n− ) n(n− ) 0 0 1 ··· 1 2 (β + n − 1) 1 (β + n) 2 n−3 2 n−2 ··· ... ..

... ··· ..

0 0 0 ··· (−1)n−1 (−1)n−1n(n + β)

Example 2. Taking

2 2 β w β w g(w) = e 2 , f (w) = f¯(w) = w and g( f¯(w)) = e 2

in extended hybrid Sheffer sequences (r) s (x, y; α), we get the extended hybrid Hermite sequences (EhHS) of νe n (β) variance β, (r) H (x, y; α). The corresponding results are given in Table4. νe n Mathematics 2019, 7, 1105 9 of 16

(β) Table 4. Results for (r) H (x, y; α). νe n

S. No. Results Expressions −ν ∂ ∂r (β) (β) I. Operational rule α − y y r {H (x)} = (r) H (x, y; α) ∂y ∂x n νe n

2 − w ∞ exwe β 2 (β) wn II. Generating function = (r) H (x, y; α) (α−yD y wr )ν ∑ νe n n! y n=0 r ˆ = − ∂ − ˆ = III. Multiplicative and M (r) H x ryDyy r−1 βDx , P (r) H Dx νe ∂α∂x νe derivative operators

∂r+1 2 (β) IV. Differential equation xD − ryD y r − βD − n (r) H (x, y; α) = 0 x y ∂α∂x x ν e n

(β) For the extended hybrid Hermite sequences (r) H (x, y; α) of degree n, the following holds: νe n

(β) (r) H (x, y; α) = 1, νe 0

( ) ( ) ( ) ( ) e r (x y ) e r (x y ) ··· e r (x y ) e r (x y ) 1 ν 1 , ; α ν 2 , ; α ν n−1 , ; α ν n , ; α

β n−1/2 β n/2 (n−1)! 2 n! 2 1 0 1 ··· n−1 ! n ! 2 2

β n−2/2 β n−1/2 (36) n−1 (n−2)! 2 n (n−1)! 2 (β) n 0 1 0 ··· ( ) ( ) (r) Hn (x, y; α) = (−1) 1 n−2 ! 1 n−1 ! . νe 2 2

β n−3/2 β n−2/2 n−1 (n−3)! 2 n (n−2)! 2 0 0 1 ··· ( ) ( ) 2 n−3 ! 2 n−2 ! 2 2

... ··· ..

0 0 0 ··· 1 0

Example 3. Taking

ew + 1 ew + 1 g(w) = , f (w) = f¯(w) = w and g( f¯(w)) = 2w 2w in extended hybrid Sheffer sequences (r) s (x, y; α), we get the extended hybrid Genocchi sequences (EhGS) νe n (r) G (x, y; α). The corresponding results are given in Table5. νe n

Table 5. Results for (r) G (x, y; α). ν e n

S. No. Results Expressions −ν ∂ ∂r I. Operational rule α − y y r {G (x)} = (r) G (x, y; α) ∂y ∂x n νe n ∞ 2w xw r −ν wn II. Generating function w e (α − yD y w ) = (r) G (x, y; α) e +1 y ∑ ν e n n! n=0 r Dx ˆ = − ∂ − e (Dx −1)−2 ˆ = III. Multiplicative and M (r) G x ryDyy r−1 Dx , P (r) G Dx νe ∂α∂x Dx (e +1) νe derivative operators

r+1 Dx ∂ e (Dx −1)−2 IV. Differential equation xD − ryD y r − − n (r) G (x, y; α) = 0 x y ∂α∂x (eDx +1) νe n Mathematics 2019, 7, 1105 10 of 16

For the extended hybrid Genocchi sequences (r) G (x, y; α) of degree n, the following holds: νe n

(r) G (x, y; α) = 1, νe 0 (r) (r) (r) (r) 1 e (x, y; α) e (x, y; α) ··· e (x, y; α) e (x, y; α) ν 1 ν 2 ν n−1 ν n

1 1 1 ··· 1 1 4 6 2n 2(n+1)

(2) 1 ··· (n−1) 1 (n) 1 (37) n 0 1 1 4 1 2(n−1) 1 2n (r) Gn(x, y; α) = (−1) . νe

n−1 1 n 1 0 0 1 ··· ( ) ( ) 2 2(n−2) 2 2(n−1)

... ··· ..

... ··· .. n 1 0 0 0 ··· 1 (n−1) 4

Example 4. Taking 2 λ −w g(w) = √ , f (w) = √ ; 1 + 1 − w2 1 + 1 − w2 −2w f¯(w) = and g( f¯(w)) = (1 + w2)λ 1 + w2 in the extended hybrid Sheffer sequences (r) s (x, y; α), we get the extended hybrid Gegenbauer sequences νe n (λ) (EhGeS) (r) C (x, y; α). The corresponding results are given in Table6. νe n

(λ) Table 6. Results for (r) C (x, y; α). νe n

S.No. Results Expressions −ν ∂ ∂r (λ) (λ) I. Operational rule α − y y r {C (x)} = (r) C (x, y; α) ∂y ∂x n νe n

−2wx exp( ) ∞ n 1+w2 (λ) w II. Generating function 2 λ −2w r ν = ∑ (r) Cn (x, y; α) (1+w ) (α−yDyy( ) ) νe n! 1+w2 n=0 √ r ( − 2 )−1/2 (1+ 1−D2 ) III. Multiplicative and Mˆ = x − ryD y ∂ − Dx λ 1√ Dx √ x , (r) C y ∂α∂xr−1 2 2 2 2 −1/2 νe (1− 1−Dx ) −(1+ 1−Dx )−Dx (1−Dx ) Pˆ = − √Dx (r) C 2 νe 1+ 1−Dx derivative operators √ r ( − 2 )−1/2 −D (1+ 1−D2 ) IV. Differential equation x − ryD y ∂ − Dx λ 1√ Dx √ x x √ − n y ∂α∂xr−1 2 2 2 2 −1/2 2 (1− 1−Dx ) (−(1+ 1−Dx )−Dx (1−Dx ) )(1+ 1−Dx ) (λ) (r) C (x, y; α) = 0 ν e n

(λ) For the extended hybrid Gegenbauer sequences (r) C (x, y; α) of degree n, the following holds: νe n

(λ) (r) C (x, y; α) = 1, νe 0

(r)( ) (r)( ) ··· (r) ( ) (r)( ) 1 νe1 x, y; α νe2 x, y; α νen−1 x, y; α νen x, y; α

λ n!λ(λ+n−1)! 1 0 ··· 0 2λ(λ+1) 2n n !λ(λ+1)...(λ+n−1) λ+ n ! 2 2

(38) n(n+3) n(n+1) 1 (λ) 0 − 2 0 ··· 0 (r) Cn (x, y; α) = (−1) 2 2 2 . νe ( − ) ( + )( + − ) 0 0 1 ··· 0 n n 1 ...3 λ 2 λ n 1 ! 4 n n−2 n+2 2 2 !(λ+2)...(λ+n−1) λ+ 2 !

... ··· ..

··· ... .. −1 n−1 0 0 0 ··· 2 0 Mathematics 2019, 7, 1105 11 of 16

In the next section, the applications of operational rules are considered.

4. Applications In order to give applications of the operational rules derived in previous sections, we use the following operation:

−ν ∂ ∂r O: Operating exp α − y ∂y y ∂xr on both sides of a given result.

We recall that the Sheffer sequences (sn(x))n∈N are deﬁned by following series representation:

n k sn(x) = ∑ bn,k x , (39) k=0 where bn,k is the (n, k) entry of the Riordan matrix ! 1 , f (t) , g( f (t)) which has the following determinant representation:

a a ··· a a k+1,k k+2,k n−1,k n,k

ak+1,k+1 ak+2,k+1 ··· an−1,k+1 an,k+1

− (−1)n k bn,k = 0 ak+2,k+2 ··· an−1,k+2 an,k+2 , (40) ak,k ... an,n

.. ··· ..

0 0 ··· an−1,n−1 an,n−1 where a is the (n, k) entry of the Riordan matrix g(t), f (t), whose inverse matrix is 1 , f (t) . n,k g( f (t))

Performing operation (O) on both sides of Equation (39) and then using appropriate operational rules, we ﬁnd the following series representation for the extended hybrid Sheffer sequences

(r) s (x, y; α): νe n n (r) (r) s (x, y; α) = b e (x, y; α). (41) νe n ∑ n,k ν k k=0

We recall that the generalized Riordan arrays for cn = 1 reduce to classical Riordan arrays and for cn = n! reduce to exponential Riordan arrays.

t t • The exponential Riordan matrix 1, t−1 with f (t) = t−1 is the Lah matrix (L(n, k))n, k∈N, whose (n, k) entry is n! n − 1 a = (−1)k = (−1)k L(n, k), (42) n,k k! n − k where L(n, k) are the Lah numbers. Mathematics 2019, 7, 1105 12 of 16

In view of Equations (40)–(42), we ﬁnd the following series representation for the sequence

(r) s (x, y; α) in terms of Lah matrix: νe n

n n (r) (r) s (x, y; α) = (−1) e (x, y; α) ν e n ∑ ν k k=0

L(k + 1, k) L(k + 2, k) ··· L(n − 1, k) L(n, k)

L(k + 1, k + 1) L(k + 2, k + 1) ··· L(n − 1, k + 1) L(n, k + 1) (43)

× . 0 L(k + 2, k + 2) ··· L(n − 1, k + 2) L(n, k + 2)

.. ··· ..

0 0 ··· L(n − 1, n − 1) L(n, n − 1)

• The exponential Riordan matrix (1, log(1 + t)) with f (t) = et − 1 is the Stirling matrix of the

ﬁrst kind (s(n, k))n, k∈N, whose (n, k) entry is

an,k = S(n, k), (44) where S(n, k) are the Stirling numbers of the second kind.

In view of Equations (40), (41) and (44), we ﬁnd the following series representation for the sequence (r) s (x, y; α) in terms of Stirling matrix of the second kind: νe n

n n−k (r) (r) s (x, y; α) = (−1) e (x, y; α) ν e n ∑ ν k k=0

S(k + 1, k) S(k + 2, k) ··· S(n − 1, k) S(n, k)

S(k + 1, k + 1) S(k + 2, k + 1) ··· S(n − 1, k + 1) S(n, k + 1) (45)

× . 0 S(k + 2, k + 2) ··· S(n − 1, k + 2) S(n, k + 2)

.. ··· ..

0 0 ··· S(n − 1, n − 1) S(n, n − 1)

• The classical Riordan matrix 1 , t with f (t) = t is the Pascal matrix (n) , 1−t t−1 1+t k n, k∈N whose (n, k) entry is − n a = (−1)n k . (46) n,k k In view of Equations (40), (41) and (46), we ﬁnd the following series representation for the sequence (r) s (x, y; α) in terms of the Pascal matrix: νe n Mathematics 2019, 7, 1105 13 of 16

n (r) (r) s (x, y; α) = e (x, y; α) νe n ∑ ν k k=0

+ + − (k 1)(k 2) ··· (n 1)(n) k k k k

k+1 k+2 n−1 n (k+ )(k+ ) ··· (k+ )(k+ ) (47) 1 1 1 1

× . 0 (k+2) ··· (n−1)( n ) k+2 k+2 k+2

.. ··· ..

.. ··· .. n−1 n 0 0 ··· (n−1)(n−1)

Next, by performing the operation (O) on both sides of Equations (5) and (9) and then use of appropriate operational rules in the resultant equations, we ﬁnd the following functional equations for the extended hybrid sequences (r) s (x, y; α) and (r) A (x, y; α): νe n νe n

n n (r) s (x + w, y; α) = (r) s (x, y; α) (r) p (w, y; α), (48) νe n ∑ νe k νe n−k k=0 k

n n (r) (r) A (x + w, y; α) = (r) A (x, y; α) e (w, y; α), (49) νe n ∑ νe k ν n−k k=0 k n Using determinant representation of binomial coefﬁcient (k) [23], the above equations can be expressed as:

n (r) s (x + w, y; α) = (r) s (x, y; α) (r) p (w, y; α) νe n ∑ νe k νe n−k k=0

+ + − (k 1)(k 2) ··· (n 1)(n) k k k k

k+1 k+2 n−1 n (k+ )(k+ ) ··· (k+ )(k+ ) (50) 1 1 1 1

× , 0 (k+2) ··· (n−1)( n ) k+2 k+2 k+2

.. ··· ..

.. ··· .. n−1 n 0 0 ··· (n−1)(n−1)

n (r) (r) A (x + w, y; α) = (r) A (x, y; α) e (w, y; α) νe n ∑ νe k ν n−k k=0

+ + − (k 1)(k 2) ··· (n 1)(n) k k k k

k+1 k+2 n−1 n (k+ )(k+ ) ··· (k+ )(k+ ) (51) 1 1 1 1

× . 0 (k+2) ··· (n−1)( n ) k+2 k+2 k+2

.. ··· ..

.. ··· .. n−1 n 0 0 ··· (n−1)(n−1) Mathematics 2019, 7, 1105 14 of 16

The other identities for the Sheffer sequences sn(x) can be recalled from [10] as:

n n sn(x) = ∑ sn−k(0) pk(x), n = 0, 1, . . . , (52) k=0 k 1 n−1 n sn(x) = pn(x) − ∑ βn−k sk(x) , n = 1, 2, . . . , (53) β0 k=0 k n n pn(x) = ∑ βn−k sk(x), n = 0, 1, . . . , (54) k=0 k n n n k sn(x) = ∑ ∑ αn−iai,k x , n = 0, 1, . . . . (55) k=0 i=k i which on performing operation (O) and then using appropriate operational rules in the resultant equations yield the following identities for the hybrid sequences (r) s (x, y; α): νe n

n n (r) s (x, y; α) = s (0) (r) p (x, y; α), n = 0, 1, . . . , (56) νe n ∑ n−k νe k k=0 k 1 n−1 n (r) s (x, y; α) = (r) p (x, y; α) − β (r) p (x, y; α) , n = 1, 2, . . . , (57) νe n νe n ∑ n−k νe k β0 k=0 k n n (r) p (x, y; α) = β (r) s (x, y; α), n = 0, 1, . . . , (58) νe n ∑ n−k νe k k=0 k n n n (r) (r) s (x, y; α) = α a e (x, y; α), n = 0, 1, . . . . (59) νe n ∑ ∑ n−i i,k ν k k=0 i=k i

Next, we consider the following results for the Appell sequences An(x) ([9], (31–32) p. 1534):

n−1 1 n n An(x) = x − ∑ βn−k Ak(x) , n = 1, 2, ··· , (60) β0 k=0 k n n n x = ∑ βn−k Ak(x), n = 0, 1, ··· . (61) k=0 k which on performing operation (O) and then using the appropriate operational relations in the resultant equations, we obtain the following identities for the sequence (r) A (x, y; α): νe n

n−1 1 (r) n (r) A (x, y; α) = e (x, y; α) − β (r) A (x, y; α) , n = 1, 2, ··· , (62) νe n ν n ∑ n−k νe k β0 k=0 k n (r) n e (x, y; α) = β (r) A (x, y; α), n = 0, 1, ··· . (63) ν n ∑ n−k νe k k=0 k Mathematics 2019, 7, 1105 15 of 16

Further, we recall the following identities for the associated Laguerre, Hermite, Gegenbauer and Genocchi sequences [6]:

n n (β+γ+1)( + ) = (β)( ) (γ) ( ) Ln u v ∑ Lk u Ln−k v , (64) k=0 k n n (β)( + ) = n−k (β)( ) Hn u v ∑ u Hk v , (65) k=0 k (k + λ) n! Γ(λ) ( ) xn = C λ (x), (66) ∑ n n−k n+k+2λ+2 k 0≤k≤n, n−k≡(mod 2) 2 ( 2 )! Γ( 2 ) n n n−k Gn(x) = ∑ Gk x , (67) k=0 k which on using the appropriate operation rules in both sides yields the following identities for the (β) (β) (λ) sequences (r) L (x, y; α), (r) H (x, y; α) and (r) C (x, y; α): νe n νe n νe n

n (β+γ) n (β) (γ) (r) L (u + v, y; α) = (r) L (u, y; α) (r) L (v, y; α), (68) νe n ∑ νe k νe n−k k=0 k n (β) n (r) (β) (r) H (u + v, y; α) = e (u, y; α) (r) H (v, y; α), (69) νe n ∑ ν n−k νe k k=0 k (r) (k + λ) n! Γ(λ) (λ) νen (x, y; α) = (r) C (x, y; α), (70) ∑ n n−k n+k+2λ+2 νe k 0≤k≤n, n−k≡(mod 2) 2 ( 2 )! Γ( 2 ) n n (r) (r) G (x, y; α) = G e (x, y; α). (71) νe n ∑ k ν n−k k=0 k

Author Contributions: Funding acquisition, M.A.; Supervision, S.K.; Writing-original draft, M.R.; Editing; K.K. Funding: This work was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under the grant no. G-363-247-1440. The authors, therefore, acknowledge with thanks DSR for technical and financial support. Acknowledgments: The authors are thankful to the Reviewer(s) for several useful comments and suggestions towards the improvement of this paper. Conflicts of Interest: The authors declare no conflict of interest.

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