2-Iterated Sheffer Polynomials

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∞ tn f(t)= f , f = 0, f 6= 0 (1.2a) n n! 0 1 nX=0 and ∞ tn g(t)= g , g 6= 0. (1.2b) n n! 0 n=0 X According to Roman [27, p.18 (Theorem 2.3.4)], the polynomial sequence sn(x) is uniquely determined by two (formal) power series given by equations (1.2a) and (1.2b). The exponential generating function of sn(x) is then given by

n 1 ¯ ∞ t ex(f(t)) = s (x) , (1.3) g(f¯(t)) n n! Xn=0 for all x in C, where f¯(t) is the compositional inverse of f(t). The sequence sn(x) in equation (1.3) is the Sheﬀer sequence for the pair (g(t),f(t)).

It has been given in [18] that the Sheﬀer polynomials sn(x) are ‘quasi-monomial’, for this see [11,29]. The associated raising and lowering operators are given by g (∂ ) 1 Mˆ = x − ′ x (1.4a) s g(∂ ) f (∂ ) x ′ x and Pˆs = f(∂x), (1.4b) ∂ respectively, where ∂x := ∂x .

Also, for an invertible series g(t) and a delta series f(t), the sequence (sn(x))n N is Sheﬀer ∈ for the pair (g(t),f(t)) if and only if

k hg(t)f(t) |sn(x)i = cnδn,k, (1.5) for all n, k ≥ 0, where δn,k is the kronecker delta. Particularly, the Sheﬀer sequence for (1,f(t)) is called the associated sequence for f(t) deﬁned by the generating function of the form:

n ¯ ∞ t exf(t) = s˜ (x) (1.6) n n! n=0 X and the Sheﬀer sequence for (g(t),t) is called the Appell sequence [3] for g(t) deﬁned by the generating function of the form [27]:

1 ∞ tn ext = A (x) . (1.7) g(t) n n! n=0 X Now, we recall some preliminaries from [32].

2 Let P be the algebra of polynomials in the variable x over K and P ⋆ be the dual vector space of all functionals on P . Let (cn)n N be a fixed sequence of nonzero constants. Then for ∈ each f(t) in F, we define a linear functional f(t) in P ⋆ is defined by

n hf(t)|x i = cnan (1.8) and deﬁne a linear operator f(t) on P is deﬁned by

n n cn n k f(t)x = akx − . (1.9) cn k Xk=0 − Now, we give the concept of Riordan arrays, which was introduced by Shapiro et. al. [23] and further studied by many authors, for example see [4, 8,15–17,24, 25].

For an invertible series g(t) and a delta series f(t), a generalized Riordan array with respect to the sequence (cn)n N is a pair (g(t),f(t)), which is an inﬁnite, lower triangular array ∈ (an,k)0 k n< according to the following rule: ≤ ≤ ∞ tn (f(t))k a = g(t) , (1.10) n,k c c n k k where the functions g(t)(f(t)) /ck are called the column generating functions of the Riordan array. Particularly, the classical Riordan arrays corresponds to the case cn = 1 and the exponential Riordan arrays corresponds to the case of cn = n!. One of the most important applications n of the theory of the Riordan arrays is to deal with summations of the form an,khk [24, 25]. k=0 P Also, for any ﬁxed sequence (cn)n N, the set of all Riordan arrays (g(t),f(t)) is a group ∈ under matrix multiplication and is called a Riordan group with respect to (cn)n N. The identity ∈ of this group is (1,t) and the inverse of the array (g(t),f(t)) is (1/g(f¯(t)), f¯(t)).

Costabile et. al. in [9] proposed a determinantal definition for the classical Bernoulli polynomials. Further, Costabile and Longo in [10] introduced the determinantal definition for the Appell sequences. Later on, by using the theory of Riordan arrays and relation between the Shef- fer sequences and Riordan arrays, the determinantal definition of Appell sequences is extended to Sheffer sequences [32]. The relations between the Sheffer sequences and Riordan arrays for the case of classical Sheffer sequences and classical Riordan arrays are given in [16] and for the case of generalized Sheffer sequences and generalized Riordan arrays are given in [15, 31].

Let (sn(x))n N be Sheﬀer for (g(t),f(t)) and suppose ∈ n n x = an,ksk(x), (1.11) Xk=0 then an,k is the (n, k) entry of the Riordan array (g(t),f(t)) [31].

3 In view of above fact, the following determinantal deﬁnition for the Sheﬀer sequences holds true [32]:

Let (sn(x))n N be Sheﬀer for (g(t),f(t)), then we have ∈ s (x)= 1 , (1.12) 0 a0,0 2 n 1 n 1 x x · · · x − x

a0,0 a1,0 a2,0 · · · an 1,0 an,0 −

( 1)n 0 a1,1 a2,1 · · · an 1,1 an,1 sn(x)= − − , a0,0a1,1...an,n

0 0 a2,2 · · · an 1,2 an,2 − . . . · · · . .

. . . · · · . .

0 0 0 · · · an 1,n 1 an,n 1 − − −

( 1)n Xn+1 = a a− ...a det , 0,0 1,1 n,n Sn (n+1) × (1.13) 2 n where Xn+1 = (1,x,x ,...,x ), Sn (n+1) = (aj 1,i 1)1 i n, 1 j n+1 and an,k is the (n, k) entry − − ≤ ≤ ≤ ≤ of the Riordan array (g(t),f(t)). ×

Also, let (sn(x))n N be the sequence of polynomials deﬁned by equations (1.12) and (1.13), ∈ where an,k is the (n, k) entry of the Riordan array (g(t),f(t)), then

n k sn(x)= bn,kx , (1.14) Xk=0 where bn,k is the (n, k) entry of the Riordan array (1/g(f¯(t)), f¯(t)) and (sn(x))n N be Sheﬀer ∈ for (g(t),f(t)) [31].

Motivated by the theory of Riordan arrays and relation between the Sheffer sequences and Riordan array, in this paper, a new family of the 2-iterated Sheffer polynomials is introduced by means of generating function. The determinantal definition of the 2-iterated Sheffer polynomials is established by using the relation between Sheffer sequences and Riordan array. The quasi-monomial properties of these polynomials are derived. The 2-iterated associated Sheffer polynomials are deduced and their properties are considered. Examples of some members belonging to these families are given.

2. 2-iterated Sheﬀer polynomials

The 2-iterated Sheﬀer polynomials (2ISP in the following) are introduced by means of generating function. Further, a determinantal deﬁnition of the 2ISP is given.

4 In order to obtain the generating function of the 2ISP, we prove the following result:

Theorem 2.1. The 2ISP are deﬁned by the following generating function:

n 1 1 ¯ ¯ ∞ [2] t exp(xf2(f1(t))) = sn (x) . (2.1) g1(f¯1(t)) g2(f¯2(f¯1(t))) n! Xn=0 (1) (2) Proof. Let sn (x) and sn (x) be Sheffer for (g1(t),f1(t)) and (g2(t),f2(t)), respectively be two different polynomials defined by the generating functions of the forms:

n 1 ¯ ∞ (1) t exp(x(f1(t))) = sn (x) (2.2a) g1(f¯1(t)) n! Xn=0 and n 1 ¯ ∞ (2) t exp(x(f2(t))) = sn (x) , (2.2b) g2(f¯2(t)) n! nX=0 respectively.

Expanding the exponential function and then replacing the powers x0, x1,...,xn by the (2) (2) (2) polynomials s0 (x), s1 (x),...,sn (x), respectively in both sides of equation (2.2a), so that we have 1 f¯ (t) (f¯ (t))n ∞ tn 1+ s(2)(x) 1 + . . . + s(2)(x) 1 = s(1)(s(2)(x)) . (2.3) g (f¯ (t)) 1 1! n n! n 1 n! 1 1 n=0 X Using equation (2.2b) with t replaced by f¯1(t) in the l.h.s. and denoting the resultant 2ISP in the r.h.s. of equation (2.3) by

[2] (1) (2) sn (x)= sn (s1 (x)), (2.4) we get assertion (2.1).

Remark 2.1. We remark that equation (2.4) is the operational correspondence between the [2] (1) 2ISP sn (x) and Sheﬀer polynomials sn (x).

Remark 2.2. We know that the Sheﬀer sequence for (g(t),t) becomes the Appell sequence 1 1 1 1 An(x). Therefore, taking f1(t) = f2(t) = t which gives = and = g1(f¯1(t)) g1(t) g2(f¯2(f¯1(t))) g2(t) in equation (2.1) yields the generating function for the 2-iterated Appell polynomials (2IAP) [2] An (x) [19].

Remark 2.3. We know that the Sheﬀer sequence for (1,f(t)) becomes the associated Sheﬀer 1 1 sequence sn(x). Therefore, taking g1(t) = g2(t) = 1 which gives = = 1 in g1(f¯1(t)) g2(f¯2(f¯1(t))) equation (2.1) yields the following consequence of Theorem 2.1: e

5 [2] Corollary 2.1. The 2-iterated associated Sheffer polynomials (2IASP) sn (x) are defined by the following generating function: e ∞ tn exp(xf¯ (f¯ (t))) = s[2](x) . (2.5) 2 1 n n! nX=0 e [2] Remark 2.4. The following operational correspondence between the 2IASP sn (x) and associated Sheffer sequences sn(x) holds:

[2] (1) (2) e e sn (x)= sn (s1 (x)). (2.6) (1) (2) It is shown in [32] that for sn (xe) and sn e(x)e be Sheffer for (g1(t),f1(t)) and (g2(t),f2(t)), respectively be two different Sheffer sequences defined by n (1) k sn (x)= bn,kx (2.7a) Xk=0 and n (2) k sn (x)= dn,kx , (2.7b) Xk=0 where bn,k is the (n, k) entry of the Riordan array (1/g1(f¯1(t)), f¯1(t)) and dn,k is the (n, k) entry (1) (2) of the Riordan array (1/g2(f¯2(t)), f¯2(t)), then the umbral composition of sn (x) and sn (x) is (2) (1) the sequence (sn (s (x)))n N defined by ∈ n n k n n n s(2)(s(1)(x)) = d s (x)= d b xj = d b xj = p xj, (2.8) n n,k k n,k k,j  n,k k,j n,j Xk=0 Xk=0 Xj=0 Xj=0 Xk=j Xj=0   1 (2) (1) where pn,j is the (n, j) entry of the Riordan array , f¯1(f¯2(t)) and sn (s (x)) g2(f¯2(t))g1(f¯1(f¯2(t))) is Sheffer for (g1(t)g2(f1(t)),f2(f1(t))).

[2] (1) Remark 2.5. We remark that, the 2ISP sn (x) are actually the composition of sn (x) and (2) sn (x) and are deﬁned by n [2] sn (x)= dn,ksk(x), (2.9) Xk=0 1 where dn,k is the is the (n, k) entry of the Riordan array , f¯1(f¯2(t)) . g2(f¯2(t))g1(f¯1(f¯2(t))) The series deﬁnition (2.9) can also be obtained by replacing the powers x1 and xk by the (2) (2) polynomials s1 (x) and sk (x), respectively, in equation (2.7b) and then using equation (2.4) in the l.h.s. of resultant equation.

[2] Now, we derive the determinantal deﬁnition for the 2ISP sn (x). For this we prove the following result:

6 [2] Theorem 2.2. The 2ISP sn (x) of degree n are deﬁned by

s[2](x)= 1 , (2.10) 0 a0,0 (2) (2) (2) (2) 1 s1 (x) s2 (x) · · · sn 1(x) sn (x) −

a0,0 a1,0 a2,0 · · · an 1,0 an,0 −

n [2] ( 1) 0 a1,1 a2,1 · · · an 1,1 an,1 sn (x)= − , a0,0a1,1...an,n −

0 0 a2,2 · · · an 1,2 an,2 − . . . · · · . .

. . . · · · . .

0 0 0 · · · an 1,n 1 an,n 1 − − −

( 1)n Sn+1(x) = a a− ...a det , 0,0 1,1 n,n Mn (n+1) × (2.11) (2) (2) where Sn+1(x) = (1,s1 (x),...,sn (x)), Mn (n+1) = (aj 1,i 1)1 i n, 1 j n+1 and an,k is the × − − ≤ ≤ ≤ ≤ (n, k) entry of the Riordan array (g1(t)g2(f1(t)),f2(f1(t))).

n (2) (2) Proof. Replacing the power x by the polynomial sn (x) in the l.h.s. and x by s1 (x) in r.h.s. of equation (1.11) and then using operational correspondence (2.4) in the r.h.s. of resultant equation, we ﬁnd n (2) [2] sn (x)= an,ksk (x), (2.12) Xk=0 where an,k is the (n, k) entry of the Riordan array (g1(t)g2(f1(t)),f2(f1(t))).

[2] The above identity leads to the following system of inﬁnite equations in the unknown sn (x) for n = 0, 1,... :

[2] a0,0s0 (x) = 1,   [2] [2] (2) a1,0s (x)+ a1,1s (x)= s (x),  0 1 1   (2.13)  [2] [2] [2] (2) a2,0s0 (x)+ a2,1s1 (x)+ a2,2s2 (x)= s2 (x), . .   [2] [2] [2] [2] (2) an,0s0 (x)+ an,1s1 (x)+ an,2s2 (x)+ . . . + an,nsn (x)= sn (x).   From ﬁrst equation of system (2.13), we get assertion (2.10). Also, the special form of system [2] (2.13) (lower triangular) allows us to work out the unknown sn (x). Operating with the ﬁrst n + 1 equations simply by applying the Cramer’s rule, we have

7 a0,0 0 0 · · · 0 1

(2) a1,0 a1,1 0 · · · 0 s1 (x)

(2) 1 a a a · · · 0 s (x) s[2](x)= 2,0 2,1 2,2 2 . (2.14) n a a . . . a 0,0 1,1 n,n . . . · · · . .

(2) an 1,0 an 1,1 an 1,2 · · · an 1,n 1 sn 1(x) − − − − − −

(2) an,0 an,1 an,2 · · · an,n 1 sn (x) −

Then, bringing (n + 1)-th column to the ﬁrst place by n transpositions of adjacent column and noting that the determinant of a square matrix is the same as that of its transpose, we obtain assertion (2.11).

Remark 2.6. We know that for (1,f(t)), the Sheffer sequences sn(x) become the associated sequences sn(x), i.e., for cn = n! in equation (1.9), (sn(x))n N is associated to f(t) and an,k ∈ is then the (n, k) entry of the Riordan array (1,f(t)) and coefficients of the associated Sheffer polynomialse obtained are of the form: tn (f(t))0 c a = = n [tn]1 = δ , (2.15) n,0 c c c n,0 n 0 0 In view of equation (2.15), the following determinantal definition of the associated Sheffer sequences sn(x) holds true [32]:

e s (x)= 1 , (2.16) 0 a0,0 2 n 1 n x x · · · x − x e

a1,1 a2,1 · · · an 1,1 an,1 − ( 1)n sn(x)= a a− ...a , 0,0 1,1 n,n 0 a2,2 · · · an 1,2 an,2 − . . · · · . .

e . . · · · . .

0 0 · · · a a n 1,n 1 n,n 1 − − −

( 1)n Sn(x) = − det , a0,0a1,1...an,n M(n 1) n ! − × e (2.17) n f where Sn(x) = (x,...,x ), M(n 1) n = (aj,i)1 i n 1, 1 j n and an,k is the (n, k) entry of the ≤ ≤ − ≤ ≤ Riordan array (1,f(t)). − × e f

8 (2) (2) Using equation (2.15) and replacing sn (x) by sn (x) in the r.h.s. of the determinantal def- [2] inition (2.10) and (2.11) of the 2ISP sn (x), we obtain the following consequence of Theorem 2.2: e [2] Corollary 2.2. The 2IASP sn (x) of degree n are deﬁned by

e s[2](x)= 1 , (2.18) 0 a0,0 (2) (2) (2) (2) s1 (x) s2 (x) · · · sn 1(x) sn (x) e −

ea1,1 ea2,1 · · · ean 1,1 ean,1 − [2] ( 1)n sn (x)= − , a0,0a1,1...an,n 0 a2,2 · · · an 1,2 an,2 − . . · · · . . e . . · · · . .

0 0 · · · an 1,n 1 an,n 1 − − −

( 1)n Sn(x) = − det , a0,0a1,1...an,n M(n 1) n ! − × e (2.19) (2) (2)f where Sn(x) = (s1 (x),..., sn (x)), M(n 1) n = (aj,i)1 i n 1, 1 j n and an,k is the (n, k) entry − × ≤ ≤ − ≤ ≤ of the Riordan array (1,f2(f1(t))). e e e f Remark 2.7. We know that for (g(t),t), the Sheﬀer sequences sn(x) become Appell sequences An(x). The coeﬃcients an,k in equation (1.13) are then the (n, k) entry of the exponential Riordan array (g(t),t) as follows:

n k t t n! n k n an,k = g(t) = [t − ]g(t)= gn k. (2.20) n! k! k! k − Therefore, using equation (2.20) in the determinantal deﬁnition (2.10) and (2.11) of the [2] [2] 2ISP sn (x) yields the determinantal deﬁnition of the 2IAP An (x), which is given in [20].

3. Quasi-monomial and other properties

[2] In this section, we frame the 2ISP sn (x) within the context of monomiality principle and [2] derive certain other properties of the 2ISP sn (x).

[2] In order to derive the multiplicative and derivative operators for the 2ISP sn (x), we prove the following result:

[2] Theorem 3.1. The 2ISP sn (x) are quasi-monomial with respect to the following multiplicative and derivative operators:

g1′ (f2(∂x)) g2′ (∂x) 1 Mˆ [2] = x − f ′ (f (∂ )) − (3.1a) s g (f (∂ )) 1 2 x g (∂ ) f (f (∂ )) f (∂ ) 1 2 x 2 x 1′ 2 x 2′ x 9 and ˆ Ps[2] = f1(f2(∂x)), (3.1b) ∂ respectively, where ∂x := ∂x .

Proof. Differentiating equation (2.1) partially with respect to t, we find ¯ ¯ ¯ g2′ (f2(f1(t))) g1′ (f1(t)) xf¯′ (f¯2(f¯1(t))) f¯′ (f¯1(t)) − f¯′ (f¯2(f¯1(t))) f¯′ (f¯1(t)) − f¯′ (f¯1(t)) 2 1 g (f¯ (f¯ (t))) 2 1 g (f¯ (t)) 1 2 2 1 1 1 n 1 1 ∞ [2] t exp(xf¯2(f¯1(t))) = s (x) , (3.2) g (f¯ (t)) g (f¯ (f¯ (t))) n+1 n! 1 1 2 2 1 n=0 X which can also be simplified as ¯ ¯ ¯ g2′ (f2(f1(t))) g1′ (f1(t)) ¯ ¯ 1 1 1 x − ¯ ¯ − ¯ f2′ (f2(f1(t))) ¯ ¯ ¯ ¯ ¯ ¯ g2(f2(f1(t))) g1(f1(t)) f ′2(f2(f1(t))) f1′ (f1(t)) g1(f1(t)) g2(f2(f1(t))) n ∞ tn exp(xf¯ (f¯ (t))) = s[2] (x) . (3.3) 2 1 n+1 n! o nX=0 g′ (t) g′ (t) Since g (t) and g (t) are invertible series of t, therefore 1 and 2 possess power series 1 2 g1(t) g2(t) [2] expansions of t. Thus, in view of the following identity for the 2ISP sn (x):

1 1 1 1 ∂ exp(xf¯ (f¯ (t))) = f¯ (f¯ (t)) exp(xf¯ (f¯ (t))) , x g (f¯ (t)) g (f¯ (f¯ (t))) 2 1 2 1 g (f¯ (t)) g (f¯ (f¯ (t))) 2 1 1 1 2 2 1 1 1 2 2 1 (3.4a) or, equivalently

1 1 1 1 f (f (∂ )) exp(xf¯ (f¯ (t))) = t exp(xf¯ (f¯ (t))) , 1 2 x g (f¯ (t)) g (f¯ (f¯ (t))) 2 1 g (f¯ (t)) g (f¯ (f¯ (t))) 2 1 1 1 2 2 1 1 1 2 2 1 (3.4b) equation (3.3) becomes

g1′ (f2(∂x)) g2′ (∂x) 1 1 1 x − f1′ (f2(∂x)) − ¯ ¯ ¯ g1(f2(∂x)) g2(∂x) f1′ (f2(∂x)) f2′ (∂x) g1(f1(t)) g2(f2(f1(t))) n ∞ tn exp(xf¯ (f¯ (t))) = s[2] (x) . (3.5) 2 1 n+1 n! n=0 o X Again, using generating function (2.1) in the l.h.s. of equation (3.5) and then rearranging the summation yields

n n ∞ g1′ (f2(∂x)) g2′ (∂x) 1 [2] t ∞ [2] t x − f ′ (f (∂ )) − s (x) = s (x) . g (f (∂ )) 1 2 x g (∂ ) f (f (∂ )) f (∂ ) n n! n+1 n! n=0 1 2 x 2 x 1′ 2 x 2′ x n=0 X X (3.6)

10 Equating the coeﬃcients of the same powers of t in both sides of the above equation, we ﬁnd

g1′ (f2(∂x)) g2′ (∂x) 1 [2] [2] x − f1′ (f2(∂x)) − sn (x) = sn+1(x), (3.7) g1(f2(∂x)) g2(∂x) f1′ (f2(∂x)) f2′ (∂x) n o [2] which in view of monomiality principle equation Mˆ {sn(x)} = sn+1(x) [18] for sn (x) yields assertion (3.1a).

In order to prove assertion (3.1b), we use generating function (2.1) in both sides of the identity (3.4b), so that we have

n n ∞ [2] t ∞ [2] t f1(f2(∂x)) sn (x) = sn 1(x) . (3.8) ( n!) − (n − 1)! nX=0 nX=1 Rearranging the summation in the l.h.s. of equation (3.8) and then equating the coeﬃcients of the same powers of t in both sides of the resultant equation, we ﬁnd

[2] [2] f1(f2(∂x)) sn (x) = n sn 1(x), n ≥ 1, (3.9) − n o [2] which in view of monomiality principle equation Pˆ{sn(x)} = n sn 1(x) [18] for sn (x) yields − assertion (3.1b).

[2] Theorem 3.2. The 2ISP sn (x) satisfy the following diﬀerential equation: x g (f (∂ )) f (f (∂ )) g (∂ ) f (f (∂ )) − 1′ 2 x 1 2 x − 2′ x 1 2 x − n s[2](x) = 0. (3.10) f (f (∂ )) g (f (∂ )) f (∂ ) g (∂ ) f (f (∂ )) f (∂ ) n 1′ 2 x 1 2 x 2′ x 2 x 1′ 2 x 2′ x

Proof. Using equations (3.1a) and (3.1b) in monomiality principle equation Mˆ Pˆ{sn(x)} = [2] nsn(x) [18] for sn (x), we get assertion (3.10).

Remark 3.1. We know that the Sheﬀer sequence for (1,f(t)) becomes the associated Sheﬀer sequence sn(x). Therefore, taking g1(t)= g2(t) = 1 which implies g1′ (t)= g2′ (t) = 0 in equations (3.1a), (3.1b) and (3.10), we get the following consequence of Theorem 3.1 and Theorem 3.2: e [2] Corollary 3.1. The 2IASP sn (x) are quasi-monomial with respect to the following multiplicative and derivative operators: ˆ x e Mse[2] = (3.11a) f1′ (f2(∂x)) f2′ (∂x) and ˆ Pse[2] = f1(f2(∂x)), (3.11b) respectively.

[2] Corollary 3.2. The 2IASP sn (x) satisfy the following differential equation: x f (f (∂ )) e 1 2 x − n s[2](x) = 0. (3.12) f (f (∂ )) f (∂ ) n 1′ 2 x 2′ x e 11 Remark 3.2. We know that for (g(t),t), the Sheffer sequences sn(x) reduce to the Appell sequences An(x). Therefore, taking f1(t)= f2(t)= t which implies f1′ (t)= f2′ (t) = 1 in equations (3.1a), (3.1b) and (3.10) yields the multiplicative and derivative operators and differential equa- [2] tion for the 2IAP An (x), which are given in [19].

[2] Now, we prove the conjugate representation of the 2ISP sn (x). For this, we prove the following result:

[2] Theorem 3.2. Let sn (x) be Sheﬀer for (g1(t)g2(f1(t)),f2(f1(t))), then the following conjugate [2] representation for the 2ISP sn (x) holds true:

n ¯ ¯ ¯ 1 ¯ ¯ k [2] h(g2(f2(t))g1(f1(f2(t))))− (f1(f2(t))) | sn(x)i sn (x)= sk(x). (3.13) ck Xk=0 [2] Proof. Since sn (x) are deﬁned by equation (2.9), where dn,k is the (n, k) entry of the Riordan 1 array , f¯1(f¯2(t)) . Then, according to the deﬁnition of Riordan arrays, we g2(f¯2(t))g1(f¯1(f¯2(t))) have tn 1 (f¯ (f¯ (t)))k d = 1 2 . (3.14) n,k c g (f¯ (t))g (f¯ (f¯ (t))) c n 2 2 1 1 2 k Simplifying the above equation yields

n 1 t 1 k d = (g (f¯ (t))g (f¯ (f¯ (t))))− ((f¯ (f¯ (t)))) n,k c c 2 2 1 1 2 1 2 k n 1 1 k = h(g2(f¯2(t))g1(f¯1(f¯2(t))))− (f¯1(f¯2(t))) | sn(x)i, (3.15) ck Now, using equation (3.15) in equation (2.9), we are led to assertion (3.13).

In the next section, examples of some members belonging to the 2ISP and 2IASP are considered.

4. Examples

(α) We recall that the Laguerre polynomials Ln (x) of order α form the Sheﬀer sequence for 1 t the pair (1 t)α+1 , t 1 [2, 22,27] are deﬁned by the generating function: − − 1 −xt ∞ exp = L(α)(x)tn, (4.1) (1 − t)α+1 1 − t n n=0 X which for α = 0 gives the generating function of the Laguerre polynomials Ln(x) as [2]:

1 −xt ∞ exp = L(α)(x)tn. (4.2) (1 − t) 1 − t n n=0 X

12 λ Also, the polynomials of the Gegenbauer case denoted by −n sn(x) considered in [32] form λ0 the Sheﬀer sequence for the pair 2 , t and are deﬁned by the generating 1+√1 t2 1+√−1 t2 − − function of the form:

∞ 2 λ λ0 2 λ −λ n (1 + t ) − (1 − 2xt + t )− = s (x)t , (4.3) n n n=0 X (λ) which for λ0 = λ gives the generating function of the Gegenbauer polynomials Cn (x) as [1,22, 26]:

2 λ ∞ (λ) n (1 − 2xt + t )− = Cn (x)t . (4.4) nX=0 We note that corresponding to each member belonging to the Sheﬀer family, there exists a new special polynomial belonging to the 2ISP family. Thus, by making suitable choice for the functions g(t) and f(t) in equation (2.1), we get the generating function for the corresponding member belonging to the 2ISP family. The other properties of these special polynomials can be obtained from the results derived in previous sections.

We consider the following examples:

1 t Example 4.1. Taking g1(t)= g2(t)= (1 t)α+1 and f1(t)= f2(t)= t 1 in the l.h.s. of generating function (2.1), we ﬁnd that the resultant− 2-iterated Laguerre polynomials− (2ILP) of order α, (α)[2] which may be denoted by Ln (x) in the r.h.s. are deﬁned by the following generating function:

1 ∞ (α)[2] n α+1 α 1 exp(xt)= Ln (x)t . (4.5) (1 − t) (1 − t)− − nX=0 (α)[2] The operational correspondence between the 2ILP of order αLn (x) and Laguerre poly- (α) nomials of order, α Ln (x) is given by:

(α)[2] (α,1) (α,2) L (x)= Ln (L1 (x)). (4.6)

(α)[2] The 2ILP of order α, Ln (x) are quasi-monomial with respect to the following multiplicative and derivative operators:

3 (α + 1) Mˆ (α)[2] = x − (α + 1)(∂ − 1) − , (4.7a) L x (∂ − 1) x ˆ PL(α)[2] = ∂x. (4.7b) (α)[2] The 2ILP of order α, Ln (x) satisfy the following diﬀerential equation:

(α + 1)∂ x∂ − (α + 1)(∂ − 1)3∂ − x − n L(α)[2](x) = 0. (4.8) x x x (∂ − 1) n x

13 (α) Consider the following series deﬁnition for the Laguerre polynomials of order α, Ln (x) [27]:

n (−1)k n + α L(α)(x)= xk. (4.9) n k! n − k Xk=0 k (α) (α) Replacing the power x by the polynomial Lk (x) in the r.h.s. and x by L1 (x) in the l.h.s. of equation (4.9) and then using equation (4.6) in the l.h.s. of resultant equation yields (α)[2] the following series deﬁnition for the 2ILP of order α, Ln (x):

n (−1)k n + α L(α)[2](x)= L(α)(x). (4.10) n k! n − k k Xk=0 k n! n+α It has been shown in [32] that for an,k = (−1) k! n k the determinantal definition of the Sheffer polynomials given by equations (1.12) and (1.13)− reduces to the determinantal definition (α) of the Laguerre polynomials of order α, Ln (x).

(α) k n! n+α Therefore, taking sn(x)= Ln (x) and an,k = (−1) k! n k in equations (2.10) and (2.11), − (α)[2] we ﬁnd that the following determinantal deﬁnition of the 2IL P of order α, Ln (x) holds true:

(α)[2] L0 (x) = 1, (4.11)

(α) (α) (α) (α) 1 L1 (x) L2 (x) . Ln 1(x) Ln (x) −

1 α + 1 (α + 2)2 . (α + n − 1)n 1 (α + n)n −

(α)[2] n(n+3) 0 −1 −2(α + 2) . −(n − 1)(α + n − 1)n 2 −n(α + n)n 1 L (x) = (−1) 2 − − , n (n 1)(n 2) n(n 1) 0 0 1 . − − (α + n − 1)n 3 − (α + n)n 2 2 − 2 − ......

0 0 0 . (−1)n 1 (−1)n 1n(n + α) − −

n(n+3) Sn+1(x) = (−1) 2 det , Mn (n+1) × (4.12) (α) (α) (α) where Sn+1(x) = (1,L1 (x),...,Ln (x)), Mn (n+1) = (aj 1,i 1)1 i n, 1 j n+1 and Ln (x) (n = − − ≤ ≤ ≤ ≤ 0, 1,...) are the Laguerre polynomials of order× α.

Remark 4.1. We remark that by taking α = 0 in the results derived above for the 2-iterated Laguerre polynomials (2ILP) of order α, we get the corresponding results for the 2-iterated [2] Laguerre polynomials (2ILP), which may be denoted by Ln (x).

14 λ0 Example 4.2. Taking g (t)= g (t)= 2 and f (t)= f (t)= t in the l.h.s. of 1 2 1+√1 t2 1 2 1+√−1 t2 − − generating function (2.1), we ﬁnd that the resultant 2-iterated polynomials of the Gegenbauer λ s[2] case (2IPoGc), which may be denoted by −n n (x) in the r.h.s. are deﬁned by the following generating function: λ 1+ t2 0 4xt(1 + t2) ∞ −λ exp = s[2](x)tn. (4.13) 1 + 6t2 + t4 1 + 6t2 + t4 n n n=0 X [2] The operational correspondence between the 2IPoGc sn (x) and polynomials of the Gegen- bauer case sn(x) is given by: s[2] s(1) s(2) n (x)= n ( 1 (x)). (4.14) λ s[2] The 2IPoGc −n n (x) are quasi-monomial with respect to the following multiplicative and derivative operators:

λ0 1 2 λ0+2 λ02 − ∂x(1 + 1 − ∂x) Mˆ s(α)[2] = x − 2 2 2 2 2 2 λ0+1 2(1 − ∂x + 1 − ∂x) 1+ 1 −p∂x + 2(1 − ∂x + 1 − ∂x) λ ∂q p p 1+ q1 − ∂2 p − 0 x x . 2 2 1 − ∂x(1 + 1 − ∂x) 2 2 2 2 2 2(1 − ∂x + 1 − ∂x) (1+ p1 − ∂x + 2(1 − ∂x + 1 − ∂x)) p 1 p q q × p p p (4.15) 2 (1 − ∂x)

p −∂x Pˆs(α)[2] = . (4.16) 2 2 2 1+ 1 − ∂x + 2(1 − ∂x + 1 − ∂x) q λ s[2] p p The 2IPoGc −n n (x) satisfy the following diﬀerential equation:

λ 2λ0 1∂ (1 + 1 − ∂2)λ0+2 x + 0 − x x 2 2 2 2 2 2 λ0+1 2(1 − ∂x + 1 − ∂x) 1+ 1 −p∂x + 2(1 − ∂x + 1 − ∂x) q λ ∂ p p q∂ (1 + 1 −p∂2) + 0 x x x 2 2 1 − ∂x(1 + 1 − ∂x) 2 2 2 2 2 2 2(1 − ∂x + 1 − ∂x) (1+ 1p− ∂x + 2(1 − ∂x + 1 − ∂x)) p p q q 1 s[2] p p p × − n n (x) = 0. (4.17) 1 − ∂2 x p It has been shown in [32] that by taking

0 if n − k is odd, k an,k = cn( 1) λ0+k λ0+n (4.18) ( c −2nλ +n n−k if n − k is even, k 0 2

λ where cn = 1/ n , the determinantal definition of the Sheffer polynomials given by equations (1.12) and (1.13) reduces to the determinantal definition of the polynomials of Gegenbauer case

15 λ s −n n(x).

(2) Therefore, taking sn (x) = snx and using equation (4.18) in equations (2.10) and (2.11), λ s[2] we ﬁnd that the following determinantal deﬁnition of the 2IPoGc −n n (x) holds true:

s[2] 0 (x) = 1, (4.19)

1 s1(x) s2(x) . sn 1(x) sn(x) −

λ n!λ0(λ0+n 1)! 1 0 0 . 0 − 2λ(λ+1) n n n 2 !λ(λ+1)...(λ+n 1) λ0+ ! 2 − 2

0 − 1 0 . 0 [2] n(n+3) n(n+1) 2 sn (x) = (−1) 2 2 2 ,

1 n(n 1)...3(λ0+2)(λ0+n 1)! 0 0 . 0 − − 4 n n−2 n+2 2 !(λ+2)...(λ+n 1) λ0+ ! 2 − 2 ...... .

n 1 0 0 0 . 1 − 0 −2

n(n+3) n(n+1) Sn+1(x) = (−1) 2 2 2 det , Mn (n+1) × (4.20) where Sn+1(x) = (1, s1(x),..., sn(x)), Mn (n+1) = (aj 1,i 1)1 i n, 1 j n+1 and sn(x) (n = − − ≤ ≤ ≤ ≤ 0, 1,...) are the polynomials of Gegenbauer× case.

Remark 4.2. We remark that, by taking λ0 = λ in the results derived above for the 2IPoGc λ s[2] −n n (x) we get the corresponding results for the 2-iterated Gegenbauer polynomials (2IGP) (λ)[2] which may be denoted by Cn (x).

Remark 4.3. It is given in [1] that, for λ = 1, the Gegenbauer polynomials becomes the Cheby- (1) shev polynomials of the second kind Un(x), i.e., Un(x) = Cn (x) and for λ = 1/2, the Gegen- (1/2) bauer polynomials becomes the Legendre polynomials Pn(x), i.e., Pn(x)= Cn (x). Therefore, taking λ = 1 and λ = 1/2 in the results of the 2-iterated Gegenbauer polynomials (2IGP) (λ)[2] Cn (x), we obtained the corresponding results of the 2-iterated Chebyshev polynomials of the [2] [2] second kind Un (x) and 2-iterated Legendre polynomials Pn (x).

Now, we proceed to introduce certain members belonging to the 2IASP. We recall that the x associated Sheﬀer family contains the falling factorials and exponential polynomials φn(x) a n as the important members.

The falling factorial x associated to f(t) = eat − 1 [27] is defined by the generating a n 16 function: n 1 ∞ x t exp(xa− log(1 + t)) = (4.21) a n n! n=0 X and the exponential polynomials φn(x) associated to f(t) = log(1 + t) [5, 27] defined by the generating function: ∞ tn exp(x(et − 1)) = φ (x) . (4.22) n n! n=0 X Also, for each member belonging to the associated Sheffer family, there exists a new special polynomial belonging to the 2IASP family. Thus, by making suitable choice for the functions f1(t) and f2(t) in equation (2.5), we get the generating function for the corresponding member belonging to the 2IASP family. The other properties of these special polynomials can be obtained from the results derived in previous sections.

We consider the following examples:

at Example 4.3. Taking f1(t)= f2(t)= e − 1 in the l.h.s. of generating function (2.5), we find x [2] that the resultant 2-iterated falling factorial (2IFF), denoted by a n in the r.h.s. are defined by the following generating function: [2] n 1 1 ∞ x t exp(xa− log(1 + a− log(1 + t))) = . (4.23) a n n! Xn=0 [2] The operational correspondence between the 2IFF x and falling factorial x is given a n a n by: (2) x [2] x (1) = a 1 . (4.24) a n a n [2] The 2IFF x are quasi-monomial with respect to the following multiplicative and deriva- a n tive operators: ˆ x M [2] = a∂ , (4.25a) x ae x a∂x a n ae ae a(ea∂x 1) ˆ [2] P x = e − − 1. (4.25b) a n x [2] The 2IFF a n satisfy the following differential equation:

a(ea∂x 1) x e − − 1 x [2] a∂x − n = 0. (4.26) aeae aea∂x ! a n Consider the following series definition for the falling factorial x [27]: a n n x = ans(n, k)xk, (4.27) a n Xk=0 17 n where s(n, k) are the Stirling numbers of the first kind defined by s(n, k) = , k, n ∈ k N, 1 ≤ k

Replacing the power xk by the polynomial x in the r.h.s. and x by x in the l.h.s. a k a 1 of equation (4.27) and then using equation (4.24) in the l.h.s. of resultant equation yields the x [2] following series definition for the 2IFF a n : n x [2] x = ans(n, k) . (4.28) a n a k Xk=0 n It has been shown in [32], that for an,k = a S(n, k), where S(n, k) are the Stirling numbers of the second kind in equations (2.16) and (2.17), the determinantal definition of the associated Sheffer polynomials reduces to the determinantal definition of falling factorials x . a n x n Therefore, taking sn(x) = a n and an,k = a S(n, k) in the r.h.s. of equations (2.18) and [2] (2.19), we find that the 2IFF x are defined by the following determinantal definition: e a n [2] x = 1, (4.29) a 0 x x · · · x x a 1 a 2 a n 1 a n − 2 n 1 n aS(1, 1) a S(2, 1) · · · a − S(n − 1, 1) a S(n, 1) [2] n+1 x ( 1) a = − n+1 , n a( 2 ) 0 a2S(2, 2) · · · an 1S(n − 1, 2) anS(n, 2) − . . · · · . .

. . · · · . .

n 1 n 0 0 · · · a − S(n − 1,n − 1) a S(n,n − 1)

( 1)n+1 Sn(x) = − det , (n+1) a 2 M(n 1) n ! − × e (4.30) x f x x where Sn(x)= a ,..., a , M(n 1) n = (aj,i)1 i n 1, 1 j n and a (n = 0, 1,...) are 1 n − × ≤ ≤ − ≤ ≤ n the falling factorials. e f

Example 4.4. Taking f1(t) = f2(t) = log(1 + t) in the l.h.s. of generating function (2.5), we [2] ﬁnd that the resultant 2-iterated exponential polynomials (2IEP) denoted by φn (x) in the r.h.s. are deﬁned by the following generating function:

n (et 1) ∞ [2] t e − − 1= φ (x) . (4.31) n n! nX=0

18 [2] The operational correspondence between the 2IEP φn (x) and exponential polynomials φn(x) is given by: [2] (1) (2) φn (x)= φn (φ1 (x)). (4.32) [2] The 2IEP φn (x) are quasi-monomial with respect to the following multiplicative and derivative operators: x Mˆ [2] = , (4.33a) φ (1 + log(1 + t))(1 + t) ˆ Pφ[2] = log(1 + log(1 + t)). (4.33b) [2] The 2IEP φn (x) satisfy the following diﬀerential equation:

x log(1 + log(1 + t)) − n φ[2](x) = 0. (4.34) (1 + log(1 + t))(1 + t) n

Consider the following series deﬁnition for the exponential polynomials φn(x) [27]:

n k φn(x)= S(n, k)x , (4.35) Xk=0 where S(n, k) are the Stirling numbers of the second kind deﬁned by

k n 1 k j k n S(n, k)= = (−1) − j . (4.36) k k! j Xj=0 k Replacing the power x by the polynomial φk(x) in the r.h.s. and x by φ1(x) in the l.h.s. of equation (4.30) and then using equation (4.35) in the l.h.s. of resultant equation yields the [2] following series deﬁnition for the 2IEP φn (x):

n [2] φn (x)= S(n, k)φk(x), (4.37) Xk=0

It has been shown in [32], that for an,k = s(n, k), where s(n, k) are the Stirling numbers of the first kind in equations (2.16) and (2.17), the determinantal definition of the associated Sheffer polynomials reduces to the determinantal definition of exponential polynomials φn(x).

Therefore, taking sn(x) = φn(x) and an,k = s(n, k) in equations (2.18) and (2.19), we [2] find that the 2-iterated exponential polynomials (2IEP) φn (x) are defined by the following determinantal definition:e

19 [2] φ0 (x) = 1, (4.37) φ1(x) φ2(x) · · · φn 1(x) φn(x) −

s(1, 1) s(2, 1) · · · s(n − 1, 1) s(n, 1)

[2] φ (x) = (−1)n+1 , n 0 s(2, 2) · · · s(n − 1, 2) s(n, 2)

. . · · · . .

0 0 · · · s(n − 1,n − 1) s(n,n − 1)

S (x) = (−1)n+1det n , M(n 1) n ! − × e (4.38) f where Sn(x) = φ1(x),...,φn(x) , M(n 1) n = (aj,i)1 i n 1, 1 j n and φn(x) (n = 0, 1,...) − × ≤ ≤ − ≤ ≤ are the exponential polynomials. e f Similar results can also be proved for certain other known members belonging to the Sheﬀer family. These members are listed in Table 1:

Table 1. Some known Sheﬀer polynomials.

S.No. A(t); H(t) g(t); f(t) Generating Functions Polynomials 2 2 t 2 n t 4 t 2xt t t I. e− ; 2t e ; 2 e − = n∞=0 Hn(x) n! Hermite polynomials Hn(x) [2] m t m P n t ( ν ) t − m t II. e− ; νt e ; ν exp(νxt t ) = n∞=0 Hn,m,ν (x) n! Generalized Hermite polynomials Hn,m,ν (x) [21] P − t 1 − t 1 1 xt L x tn III. (1 )− ; (1 )− (1 t) exp t 1 = n∞=0 n( ) Laguerre polynomials − − ! t t P t 1 t 1 n!Ln(x) [2] − − t t 1+t 2 e 1 t 1+t x tn IV. ; ln ; − = ∞ Pn(x) Pidduck polynomials 1 t 1 t et 1 et+1 1 t 1 t n=0 n! − − − − − Pn(x) [7, 13] P n βt − t − β − t (β) t V. e ; 1 e (1 t)− ; exp βt + x(1 e ) = n∞=0 an (x) n! Actuarial polynomials (β) ln(1 − t) P an (x) [7] x n t t − t t t VI. e− ; exp a(e 1) ; e− 1+ a = n∞=0 cn(x ; a) n! Poisson-Charlier polynomials t t− ln 1+ a a(e 1) P cn(x ; a) [12, 14, 30] µ µ µ VII. 1+(1+ t)λ − ; 1+ eλt ; 1+(1+ t)λ − (1 + t)x Peters polynomials n t et − s x λ, µ t s x λ, µ ln(1 + ) 1 = n∞=0 n( ; ) n! n( ; ) [7] t t t t x tn VIII. ;ln(1 + t) ; e − 1 (1 + t) = ∞ bn(x) Bernoulli polynomials ln(1+t) et 1 ln(1+P t) n=0 n! − of the second kind bn(x) [14] P n 2 1 t t − 2 x t IX. 2+t ; ln(1 + t) 2 (1 + e ); e 1 2+t (1 + t) = n∞=0 rn(x) n! Related polynomials r (x) [14] P n 1 1 tn X. ; arctan(t) sect; tan t exp(x arctan(t)) = n∞=0 Rn(x) n! Hahn polynomials 1+t2 1+t2 q q P Rn(x) [6]

1 1 1+t 2 a 1 XI. (1 − 4t)− 2 ; (1 − 4t)− 2 − Shively’s pseudo-Laguerre (1 t)a 1+√1 4t − 2 − 2 a 1 1 1 1+t 4xt n × − ; − × exp − = ∞ Rn(a,x)t polynomials Rn(a,x) [22] 1+√1 4t 4 4 1 t (1+√1 4t ) 2 n=0 − − ! − ! 4t − P (1+√1 4t ) 2 −

20 1+α+β 2+α+β 1+α+β , ; 2xt 1 2 − 1 α β 2 2 (1 t)2 XII. 1+α+β ; ; (1 t)− − − 2F1 The polynomials of (1 t) 1+√1+2t " 1+ α − # − 2t t ∞ tn = Jn(x) the Jacobi case [26] (1 t)2 1+t+√1+2t cn − n=0 2 P 2 (1 t ) 1 1 t ∞ − nt x tn XIII. (1+− t) ; ; − 2 = ( 1) n( ) The polynomials of 1 t2 1 2xt+t n=0 − − 2t q t P − 2 − the Chebyshev case [26] (1+t) 1+ 1 t2 − q (α)[2] [2] Now, we proceed to plot the graphs related to the 2ILP of order, αLn (x), 2ILP Ln (x), [2] [2] (α) 2IFF (x)n and 2IEP φn (x) for n = 4. For this, we need the first few values of Ln (x), Ln(x), (α) (x)n and φn(x). We give the list of first five Ln (x), Ln(x), (x)n and φn(x) in Table 2.

(α) Table 2. First ﬁve Ln (x), Ln(x), (x)n and φn(x).

n 0 1 2 3 4 L(α) x α − x 1 α α 1 α α α 1 α α α α − x − 1 α α α − x x n ( ) 1 1+ 2 (1 + )(2 + ) 6 (1 + )(2 + )(3 + ) 24 (1 + )(2 + )(3 + )(7 + ) 8 (2 + )(3 + )(7 + ) − α x 1 x2 − 1 α α x 1 α α − x x2 − 1 α − x x3 − 1 α α α (2 + ) + 2 2 (2 + )(3 + ) 8 (3 + )(7 + ) 24 (7 + ) 8 (1 + )(2 + )(3 + ) 1 α x2 − 1 x3 1 α α x − 1 α x + 2 (3 + ) 6 + 24 (2 + )(3 + ) 8 (3 + ) − 1 2 − 1 − 3 2 − 1 4 − 3 2 − Ln(x) 1 1 x 2 (x 4x + 2) 6 ( x + 9x 18x + 6) 24 (x 16x + 72x 96x + 24) 2 3 2 4 3 2 (x)n 0 x x − x x − 3x + 2x x − 6x + 11x − 6x 2 3 2 4 3 2 φn(x) 1 x x + x x + 3x + x x + 6x + 7x + x

In view of equations (4.10), (4.28), (4.37) and Table 2, we find the first few values of (3)[2] [2] [2] [2] (3)[2] [2] [2] Ln (x), Ln (x), (x)n and φn (x). We present the values of first five Ln (x), Ln (x), (x)n [2] and φn (x) in Table 3.

(α)[2] [2] [2] [2] Table 3. First ﬁve Ln (x), Ln (x), (x)n and φn (x).

n 0 1 2 3 4 2 3 4 3 2 (3)[2] x 5x − 11x − 2 − 5x − x 7x 35x − 35x − 175 L n (x) 1 x 4 + 2 5 36 x 2 30 576 + 48 + 16 24 8 2 3 2 4 2 L[2] x x x x − 1 x x x − 13 x 5x − x − 15 n ( ) 1 4 + 2 36 + 2 + 2 6 576 + 8 6 24 [2] 2 3 2 4 3 2 (x)n 0 x x − 2x x − 6x + 7x x − 12x + 40x − 35x [2] 2 3 2 4 3 2 φn (x) 1 x x + 2x x + 6x + 5x x + 12x + 32x + 15x

From Table 2, we get the following graphs:

Graph of L(3)[2](x) Graph of L[2](x) 4 4 20 12

15 10 10 8 5

0 6

−5 4

−10 2 −15 0 −20

−25 −2 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4

21 Graph of (x)[2] Graph of φ[2](x) 4 4 2000 1600

1400

1500 1200

1000 1000 800

600 500 400

200 0

−500 −200 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4

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