# General-Appell Polynomials within the Context of Monomiality Principle

Authors: Subuhi Khan; Nusrat Raza
Journal: International Journal of Analysis
 (2013)

---

## Abstract

A general class of the 2-variable polynomials is considered, and its properties are derived. Further, these polynomials are used to introduce the 2-variable general-Appell polynomials (2VgAP). The generating function for the 2VgAP is derived, and a correspondence between these polynomials and the Appell polynomials is established. The differential equation, recurrence relations, and other properties for the 2VgAP are obtained within the context of the monomiality principle. This paper is the first attempt in the direction of introducing a new family of special polynomials, which includes many other new special polynomial families as its particular cases.


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## Body

## 1. Introduction and Preliminaries

The Appell polynomials are very often found in different applications in pure and applied mathematics. The Appell polynomials [1] may be defined by either of the following equivalent conditions:  {An(x)}(n∈ℕ0) is an Appell set (An being of degree exactly n) if either,(i)
(d/dx)An(x)=nAn-1(x)  (n∈ℕ0) or(ii)
there exists an exponential generating function of the form(1)A(t)exp(xt)=∑n=0∞‍An(x)tnn!,
where A(t) has (at least the formal) expansion:
(2)A(t)=∑n=0∞‍Antnn!(A0≠0).Roman [2] characterized Appell sequences in several ways. Properties of Appell sequences are naturally handled within the framework of modern classical umbral calculus by Roman [2]. We recall the following result [2, Theorem 2.5.3], which can be viewed as an alternate definition of Appell sequences.The sequenceAn(x) is Appell for g(t), if and only if
(3)1g(t)exp(xt)=∑n=0∞‍An(x)tnn!,
where
(4)g(t)=∑n=0∞‍gntnn!(g0≠0).In view of (1) and (3), we have
(5)A(t)=1g(t).The Appell class contains important sequences such as the Bernoulli and Euler polynomials and their generalized forms. Some known Appell polynomials are listed in Table1.Table 1
List of some Appell polynomials.
S. No.
g
(
t
); A(t)
Generating functions
Polynomials
(I)
g
(
t
)
=
(
e
t
-
1
)
t; A(t)=t(et-1)
t
(
e
t
-
1
)
e
x
t
=
∑
n
=
0
∞
B
n
(
x
)
t
n
n
!
The Bernoulli polynomials [3]
(II)
g
(
t
)
=
(
e
t
+
1
)
2; A(t)=2(et+1)
2
(
e
t
+
1
)
e
x
t
=
∑
n
=
0
∞
E
n
(
x
)
t
n
n
!
The Euler polynomials [3]
(III)
g
(
t
)
=
(
e
t
-
1
)
α
t
α; A(t)=tα(et-1)α
t
α
(
e
t
-
1
)
α
e
x
t
=
∑
n
=
0
∞
B
n
(
α
)
(
x
)
t
n
n
!
The generalized Bernoulli polynomials [4]
(IV)
g
(
t
)
=
(
e
t
+
1
)
α
2
α; A(t)=2α(et+1)α
2
α
(
e
t
+
1
)
α
e
x
t
=
∑
n
=
0
∞
E
n
(
α
)
(
x
)
t
n
n
!
The generalized Euler polynomials [4]
(V)
g
(
t
)
=
(
e
α
1
t
-
1
)
(
e
α
2
t
-
1
)
⋯
(
e
α
m
t
-
1
)
α
1
α
2
⋯
α
m
t
m;
α
1
α
2
⋯
α
m
t
m
(
e
α
1
t
-
1
)
(
e
α
2
t
-
1
)
⋯
(
e
α
m
t
-
1
)
e
x
t
=
∑
n
=
0
∞
B
n
(
m
)
(
x
∣
α
1
,
α
2
,
…
,
α
m
)
t
n
n
!
The generalized Bernoulli polynomials of orderm [5]
A
(
t
)
=
α
1
α
2
⋯
α
m
t
m
(
e
α
1
t
-
1
)
(
e
α
2
t
-
1
)
⋯
(
e
α
m
t
-
1
)
(VI)
g
(
t
)
=
(
e
α
1
t
+
1
)
(
e
α
2
t
+
1
)
⋯
(
e
α
m
t
+
1
)
2
m;
2
m
(
e
α
1
t
+
1
)
(
e
α
2
t
+
1
)
⋯
(
e
α
m
t
+
1
)
e
x
t
=
∑
n
=
0
∞
E
n
(
m
)
(
x
∣
α
1
,
α
2
,
…
,
α
m
)
t
n
n
!
The generalized Euler polynomials of orderm [5]
A
(
t
)
=
2
m
(
e
α
1
t
+
1
)
(
e
α
2
t
+
1
)
⋯
(
e
α
m
t
+
1
)
(VII)
g
(
t
)
=
e
t
-
∑
h
=
0
m
-
1
‍
(
t
h
/
h
!
)
t
m; A(t)=tmet-∑h=0m-1‍(th/h!)
t
m
e
t
-
∑
h
=
0
m
-
1
‍
(
t
h
/
h
!
)
e
x
t
=
∑
n
=
0
∞
B
n
[
m
-
1
]
(
x
)
t
n
n
!
The new generalized Bernoulli polynomials [6]
(VIII)
g
(
t
)
=
(
t
λ
e
t
-
1
)
-
α; A(t)=(tλet-1)α
(
t
λ
e
t
-
1
)
α
e
x
t
=
∑
n
=
0
∞
𝔅
n
(
α
)
(
x
;
λ
)
t
n
n
!
The Apostol-Bernoulli polynomials of orderα [7]
(IX)
g
(
t
)
=
λ
e
t
-
1
t; A(t)=tλet-1
t
λ
e
t
-
1
e
x
t
=
∑
n
=
0
∞
𝔅
n
(
x
;
λ
)
t
n
n
!
The Apostol-Bernoulli  polynomials [7, 8]
(X)
g
(
t
)
=
(
2
λ
e
t
+
1
)
-
α; A(t)=(2λet+1)α
(
2
λ
e
t
+
1
)
α
e
x
t
=
∑
n
=
0
∞
ℰ
n
(
α
)
(
x
;
λ
)
t
n
n
!, |t+logλ|<π;1α∶=1
The Apostol-Euler polynomials of orderα [7, 9]
(XI)
g
(
t
)
=
λ
e
t
+
1
2; A(t)=2λet+1
2
λ
e
t
+
1
e
x
t
=
∑
n
=
0
∞
ℰ
n
(
x
;
λ
)
t
n
n
!,|t+logλ|<π
The Apostol-Euler polynomials [7–9]
(XII)
g
(
t
)
=
e
-
(
ξ
0
+
ξ
1
t
+
ξ
2
t
2
+
⋯
+
ξ
r
+
1
t
r
+
1
); A(t)=e(ξ0+ξ1t+ξ2t2+⋯+ξr+1tr+1); ξr+1≠0
e
(
ξ
0
+
ξ
1
t
+
ξ
2
t
2
+
⋯
+
ξ
r
+
1
t
r
+
1
)
e
x
t
=
∑
n
=
0
∞
A
n
(
x
)
t
n
n
!
The generalized Gould-Hopper polynomials [10] (for r=1, the Hermite polynomials Hn(x) [11] and for r=2, classical 2-orthogonal polynomials)
(XIII)
g
(
t
)
=
(
1
-
t
)
m
+
1; A(t)=1(1-t)m+1
1
(
1
-
t
)
m
+
1
e
x
t
=
∑
n
=
0
∞
G
n
(
m
)
(
x
)
t
n
The Miller-Lee polynomials [11, 12] (for m=0, the truncated exponential polynomials en(x) [11] and for m=β-1, the modified Laguerre polynomials fn(β)(x) [13])
(XIV)
g
(
t
)
=
(
e
t
+
1
)
2
t; A(t)=2t(et+1)
2
t
(
e
t
+
1
)
e
x
t
=
∑
n
=
0
∞
G
n
(
x
)
t
n
n
!
The Genocchi polynomials [14]We recall that, according to the monomiality principle [15, 16], a polynomial set {pn(x)}n∈ℕ is “quasimonomial”, provided there exist two operators M^ and P^ playing, respectively, the role of multiplicative and derivative operators, for the family of polynomials. These operators satisfy the following identities, for all n∈ℕ:
(6)M^{pn(x)}=pn+1(x),(7)P^{pn(x)}=npn-1(x).The operatorsM^ and P^ also satisfy the commutation relation
(8)[P^,M^]=P^M^-M^P^=1^
and thus display the Weyl group structure. If the considered polynomial set {pn(x)}n∈ℕ is quasimonomial, its properties can easily be derived from those of the M^ and P^ operators. In fact,(i)
Combining the recurrences (6) and (7), we have
(9)M^P^{pn(x)}=npn(x)which can be interpreted as the differential equation satisfied bypn(x), if M^ and P^ have a differential realization.(ii)
Assuming here and in the sequelp0(x)=1, then pn(x) can be explicitly constructed as
(10)pn(x)=M^n{p0(x)}=M^n{1}which yields the series definition forpn(x).(iii)
Identity (10) implies that the exponential generating function of pn(x) can be given in the form
(11)exp(tM^){1}=∑n=0∞‍pn(x)tnn!(|t|<∞).We note that the Appell polynomialsAn(x) are quasimonomial with respect to the following multiplicative and derivative operators: (12a)M^A=x+A′(Dx)A(Dx)
or, equivalently,
(12b)M^A=x-g′(Dx)g(Dx),(13)P^A=Dx,
respectively.The special polynomials of two variables are useful from the point of view of applications in physics. Also, these polynomials allow the derivation of a number of useful identities in a fairly straightforward way and help in introducing new families of special polynomials. For example, Bretti et al. [17] introduced general classes of two variables Appell polynomials by using properties of an iterated isomorphism, related to the Laguerre-type exponentials.We consider the 2-variable general polynomial (2VgP) familypn(x,y) defined by the generating function
(14)extϕ(y,t)=∑n=0∞‍pn(x,y)tnn!(p0(x,y)=1),
where ϕ(y,t) has (at least the formal) series expansion
(15)ϕ(y,t)=∑n=0∞‍ϕn(y)tnn!(ϕ0(y)≠0).We recall that the 2-variable Hermite Kampé de Fériet polynomials (2VHKdFP)Hn(x,y) [18], the Gould-Hopper polynomials (GHP) Hn(m)(x,y) [19], and the Hermite-Appell polynomials (HAP) HAn(x,y) [20] are defined by the generating functions
(16)ext+yt2=∑n=0∞‍Hn(x,y)tnn!,(17)ext+ytm=∑n=0∞‍Hn(m)(x,y)tnn!,(18)A(t)ext+yt2=∑n=0∞‍AHn(x,y)tnn!,
respectively. Thus, in view of generating functions (14), (16), (17), and (18), we note that the 2VHKdFP Hn(x,y), the GHP Hn(m)(x,y), and the HAP HAn(x,y) belong to 2VgP family.In this paper, operational methods are used to introduce certain new families of special polynomials related to the Appell polynomials. In Section2, some results for the 2-variable general polynomials (2VgP) pn(x,y) are derived. Further, the 2-variable general-Appell polynomials (2VgAP) pAn(x,y) are introduced and framed within the context of monomiality principle. In Section 3, the Gould-Hopper-Appell polynomials (GHAP) HAn(m)(x,y) are considered, and their properties are established. Some members belonging to the Gould-Hopper-Appell polynomial family are given.


## 2. 2-Variable General-Appell Polynomials

In order to introduce the 2-variable general-Appell polynomials (2VgAP), we need to establish certain results for the 2VgPpn(x,y). Therefore, first we prove the following results for the 2VgP pn(x,y).Lemma 1.
The 2VgPpn(x,y) defined by generating function (14), where ϕ(y,t) is given by (15), are quasimonomial under the action of the following multiplicative and derivative operators:
(19)M^p=x+ϕ′(y,Dx)ϕ(y,Dx)(ϕ′(x,t):=∂∂tϕ(x,t)),(20)P^p=Dx,
respectively.Proof.
Differentiating (14) partially with respect to t, we have
(21)(x+ϕ′(y,t)ϕ(y,t)){extϕ(y,t)}=∑n=0∞‍pn+1(x,y)tnn!.
Ifϕ(y,t) is an invertible series and ϕ′(y,t)/ϕ(y,t) has Taylor's series expansion in powers of t, then in view of the identity
(22)Dx{extϕ(y,t)}=t(extϕ(y,t))
we can write
(23)ϕ′(y,Dx)ϕ(y,Dx){extϕ(y,t)}=ϕ′(y,t)ϕ(y,t)(extϕ(y,t)).
Now, using (23) in the l.h.s. of (21), we find
(24)(x+ϕ′(y,Dx)ϕ(y,Dx)){extϕ(y,t)}=∑n=0∞‍pn+1(x,y)tnn!.
Making use of generating function (14) in the l.h.s. of the above equation, we have
(25)(x+ϕ′(y,Dx)ϕ(y,Dx)){∑n=0∞‍pn(x,y)tnn!}=∑n=0∞‍pn+1(x,y)tnn!
which, on equating the coefficients of like powers of t in both sides, gives
(26)(x+ϕ′(y,Dx)ϕ(y,Dx)){pn(x,y)}=pn+1(x,y).
Thus, in view of monomiality principle equation (6), the above equation yields assertion (19) of Lemma 1. Again, using identity (22) in (14), we have
(27)Dx{∑n=0∞‍pn(x,y)tnn!}=∑n=1∞‍pn-1(x,y)tn(n-1)!.
Equating the coefficients of like powers oft in both sides of (27), we find
(28)Dx{pn(x,y)}=npn-1(x,y)(n≥1)
which in view of monomiality principle equation (7) yields assertion (20) of Lemma 1.Remark 2.
The operators given by (19) and (20) satisfy commutation relation (8). Also, using expressions (19) and (20) in (9), we get the following differential equation satisfied by 2VgP pn(x,y):
(29)(xDx+ϕ′(y,Dx)ϕ(y,Dx)Dx-n)pn(x,y)=0.Remark 3.
Sincep0(x,y)=1, therefore, in view of monomiality principle equation (10), we have
(30)pn(x,y)=(x+ϕ′(y,Dx)ϕ(y,Dx))n{1}(p0(x,y)=1).
Also, in view of (11), (14), and (19), we have
(31)exp(M^pt){1}=extϕ(y,t)=∑n=0∞‍pn(x,y)tnn!.
Now, we proceed to introduce the 2-variable general-Appell polynomials (2VgAP). In order to derive the generating functions for the 2VgAP, we take the 2VgPpn(x,y) as the base in the Appell polynomials generating function (1). Thus, replacing x by the multiplicative operator M^p of the 2VgP pn(x,y) in the l.h.s. of (1) and denoting the resultant 2VgAP by pAn(x,y), we have
(32)A(t)e(M^pt)=∑n=0∞‍Apn(x,y)tnn!.
Now, using (31) in the exponential term in the l.h.s. of (32), we get the generating function for pAn(x,y) as
(33)A(t)extϕ(y,t)=∑n=0∞‍Apn(x,y)tnn!.
In view of (5), generating function (33) can be expressed equivalently as
(34)1g(t)extϕ(y,t)=∑n=0∞‍Apn(x,y)tnn!.
Now, we frame the 2VgAPpAn(x,y) within the context of monomiality principle formalism. We prove the following results.Theorem 4.
The 2VgAPpAn(x,y) are quasimonomial with respect to the following multiplicative and derivative operators: (35a)M^pA=x+ϕ′(y,Dx)ϕ(y,Dx)+A′(Dx)A(Dx)
or, equivalently,
(35b)M^pA=x+ϕ′(y,Dx)ϕ(y,Dx)-g′(Dx)g(Dx),(36)P^pA=Dx,
respectively.Proof.
Differentiating (33) partially with respect to t, we find
(37)(x+ϕ′(y,t)ϕ(y,t)+A′(t)A(t))A(t)extϕ(y,t)=∑n=0∞‍Apn+1(x,y)tnn!.
SinceA(t) and ϕ(y,t) are invertible series of t, therefore, A′(t)/A(t) and ϕ′(y,t)/ϕ(y,t) possess power series expansions of t. Thus, in view of identity (22), we have
(38)(x+ϕ′(y,Dx)ϕ(y,Dx)+A′(Dx)A(Dx)){A(t)extϕ(y,t)}=∑n=0∞‍Apn+1(x,y)tnn!
which, on using generating function (33), becomes
(39)(x+ϕ′(y,Dx)ϕ(y,Dx)+A′(Dx)A(Dx)){∑n=0∞‍Apn(x,y)tnn!}=∑n=0∞‍Apn+1(x,y)tnn!
or, equivalently,
(40)∑n=0∞‍(x+ϕ′(y,Dx)ϕ(y,Dx)+A′(Dx)A(Dx)){Apn(x,y)}tnn!=∑n=0∞‍Apn+1(x,y)tnn!.
Now, equating the coefficients of like powers oft in the above equation, we find
(41)(x+ϕ′(y,Dx)ϕ(y,Dx)+A′(Dx)A(Dx)){Apn(x,y)}=Apn+1(x,y)
which in view of (6) yields assertion (35a) of Theorem 4. Also, in view of relation (5), assertion (35a) can be expressed equivalently as (35b).
Again, in view of identity (22), we have
(42)Dx{A(t)extϕ(y,t)}=tA(t)extϕ(y,t)
which on using generating function (33) becomes
(43)Dx{∑n=0∞‍Apn(x,y)tnn!}=∑n=1∞‍Apn-1(x,y)tn(n-1)!.
Equating the coefficients of like powers oft in the above equation, we find
(44)Dx{Apn(x,y)}=nApn-1(x,y)(n≥1)
which, in view of (7), yields assertion (36) of Theorem 4.Theorem 5.
The 2VgAPApn(x,y) satisfy the following differential equations(45a)(xDx+ϕ′(y,Dx)ϕ(y,Dx)Dx+A′(Dx)A(Dx)Dx-n)Apn(x,y)=0
or, equivalently,
(45b)(xDx+ϕ′(y,Dx)ϕ(y,Dx)Dx-g′(Dx)g(Dx)Dx-n)Apn(x,y)=0.Proof.
Using (35a) and (36) in (9), we get assertion (45a). Further, using (35b) and (36) in (9), we get assertion (45b).Note 1.
With the help of the results derived above and by takingA(t) (or g(t)) of the Appell polynomials listed in Table 1, we can derive the generating function and other results for the members belonging to 2VgAP family.


## 3. Examples

We consider examples of certain members belonging to the 2VgAP family.Takingϕ(y,t)=eytm (that is, when the 2VgP pn(x,y) reduces to the GHP Hn(m)(x,y)) in generating function (33), we find that the Gould-Hopper-Appell polynomials (GHAP) HAn(m)(x,y) are defined by the following generating function:
(46)A(t)e(xt+ytm)=∑n=0∞‍AHn(m)(x,y)tnn!
or, equivalently,
(47)1g(t)e(xt+ytm)=∑n=0∞‍AHn(m)(x,y)tnn!.Using (1) in (46) (or (3) in (47)), we get the following series definition for HAn(m)(x,y) in terms of the Appell polynomials An(x):
(48)HAn(m)(x,y)=n!∑r=0[n/m]‍An-mr(x)yr(n-mr)!r!.In view of (35a), (35b), and (36), we note that the GHAP HAn(m)(x,y) are quasimonomial under the action of the following multiplicative and derivative operators:(49a)M^H(m)A=x+myDxm-1+A′(Dx)A(Dx)
or, equivalently,
(49b)M^H(m)A=x+myDxm-1-g′(Dx)g(Dx),(50)P^H(m)A=Dx,respectively. Also, in view of (45a) and (45b), we find that the GHAP HAn(m)(x,y) satisfy the following differential equation:(51a)(xDx+myDxm+A′(Dx)A(Dx)Dx-n)AHn(m)(x,y)=0
or, equivalently,
(51b)(xDx+myDxm-g′(Dx)g(Dx)Dx-n)AHn(m)(x,y)=0.Remark 6.
In view of (16) and (17), we note that, for m=2, the GHAP HAn(m)(x,y) reduce to the Hermite-Appell polynomials (HAP) HAn(x,y). Therefore, taking m=2 in (46), (47), (48), (49a), (49b), (50), (51a), and (51b), we get the corresponding results for the HAP HAn(x,y).Remark 7.
In view of (16), we note that the 2VHKdFP Hn(x,y) are related to the classical Hermite polynomials Hn(x) [11] or Hen(x) as
(52)Hn(2x,-1)=Hn(x),Hn(x,-12)=Hen(x).
Therefore, taking y=-1 and replacing x by 2x (or taking y=-1/2) in (46)–(51b), we get the corresponding results for the classical Hermite-Appell polynomials HAn(x) (or HeAn(x)).There are several members belonging to 2VgP family. Thus the results for the corresponding 2VgAP can be obtained by taking other examples. We present the list of some members belonging to the GHAP family in Table2.Table 2
List of some Gould-Hopper-Appell polynomialsHAn(m)(x,y).
S. No.
g
(
t
); A(t)
M
^
H
(
m
)
A; P^H(m)A
Generating functions
Polynomials
(I)
(
e
t
-
1
)
t; t(et-1)
x
+
m
y
D
x
m
-
1
+
(
(
1
-
D
x
)
e
D
x
-
1
)
D
x
(
e
D
x
-
1
); Dx
t
(
e
t
-
1
)
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
B
n
(
m
)
(
x
,
y
)
t
n
n
!
The Gould-Hopper-Bernoulli polynomials
(II)
(
e
t
+
1
)
2; 2(et+1)
x
+
m
y
D
x
m
-
1
-
e
D
x
(
e
D
x
+
1
); Dx
2
(
e
t
+
1
)
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
E
n
(
m
)
(
x
,
y
)
t
n
n
!
The Gould Hopper-Eulerpolynomials
(III)
(
e
t
-
1
)
α
t
α; tα(et-1)α
x
+
m
y
D
x
m
-
1
+
α
(
(
1
-
D
x
)
e
D
x
-
1
)
D
x
(
e
D
x
-
1
); Dx
t
α
(
e
t
-
1
)
α
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
B
n
(
m
,
α
)
(
x
,
y
)
t
n
n
!
The Gould-Hopper-generalized Bernoulli polynomials
(IV)
(
e
t
+
1
)
α
2
α; 2α(et+1)α
x
+
m
y
D
x
m
-
1
-
α
e
D
x
(
e
D
x
+
1
); Dx
2
α
(
e
t
+
1
)
α
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
E
n
(
m
,
α
)
(
x
,
y
)
t
n
n
!
The Gould-Hopper-generalizedEuler polynomials
(V)
∏
h
=
1
s
(
e
α
h
t
-
1
)
α
h
t
s;
x
+
m
y
D
x
m
-
1
+
m
D
x
-
1
-
∑
r
=
1
m
α
r
e
α
r
D
x
(
e
α
r
D
x
-
1
); Dx
α
1
⋯
α
s
t
s
(
e
α
1
t
-
1
)
⋯
(
e
α
s
t
-
1
)
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
B
n
(
m
,
s
)
(
x
,
y
∣
α
1
,
…
,
α
s
)
t
n
n
!
The Gould-Hopper-generalized Bernoulli polynomials of orders
∏
h
=
1
s
α
h
(
e
α
h
t
-
1
)
t
s
(VI)
∏
h
=
1
s
(
e
α
h
t
+
1
)
2
s;
x
+
m
y
D
x
m
-
1
-
∑
r
=
1
m
α
r
e
α
r
D
x; Dx
2
s
(
e
α
1
t
+
1
)
⋯
(
e
α
s
t
+
1
)
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
E
n
(
m
,
s
)
(
x
,
y
∣
α
1
,
…
,
α
s
)
t
n
n
!
The Gould-Hopper-generalizedEuler polynomials of order s
∏
h
=
1
s
2
s
(
e
α
h
t
+
1
)
(VII)
e
t
-
∑
h
=
0
s
-
1
‍
(
t
h
/
h
!
)
t
s;
x
+
m
y
D
x
m
-
1
+
m
D
x
-
1
-
1
-
D
x
m
-
1
/
(
m
-
1
)
!
(
e
D
x
-
∑
h
=
0
m
-
1
‍
t
h
/
h
!
); Dx
t
s
e
t
-
∑
h
=
0
s
-
1
‍
(
t
h
/
h
!
)
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
B
n
(
m
,
[
s
-
1
]
)
(
x
,
y
)
t
n
n
!
The Gould-Hopper newgeneralized Bernoulli polynomials
t
s
e
t
-
∑
h
=
0
s
-
1
‍
(
t
h
/
h
!
)
(VIII)
(
t
λ
e
t
-
1
)
-
α; (tλet-1)α
x
+
m
y
D
x
m
-
1
+
λ
e
D
x
(
α
-
D
x
)
-
α
D
x
(
λ
e
D
x
-
1
); Dx
(
t
λ
e
t
-
1
)
α
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
𝔅
n
(
m
,
α
)
(
x
,
y
;
λ
)
t
n
n
!
The Gould-Hopper-Apostol-Bernoulli polynomials of orderα
(IX)
λ
e
t
-
1
t; tλet-1
x
+
m
y
D
x
m
-
1
+
λ
e
D
x
(
1
-
D
x
)
-
1
D
x
(
λ
e
D
x
-
1
); Dx
t
λ
e
t
-
1
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
𝔅
n
(
m
)
(
x
,
y
;
λ
)
t
n
n
!
The Gould-Hopper-Apostol-Bernoulli polynomials
(X)
(
2
λ
e
t
+
1
)
-
α; (2λet+1)α
x
+
m
y
D
x
m
-
1
-
λ
α
e
D
x
(
λ
e
D
x
+
1
);           Dx
(
2
λ
e
t
+
1
)
α
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
ℰ
n
(
m
,
α
)
(
x
,
y
;
λ
)
t
n
n
!,|t+logλ|<π;1α∶=1
The Gould-Hopper-Apostol-Euler polynomials of orderα
(XI)
λ
e
t
+
1
2; 2λet+1
x
+
m
y
D
x
m
-
1
-
λ
e
D
x
(
λ
e
D
x
+
1
); Dx
2
λ
e
t
+
1
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
ℰ
n
(
m
)
(
x
,
y
;
λ
)
t
n
n
!
,
|
t
+
log
λ
|
<
π
The Gould-Hopper-Apostol-Euler polynomials
(XII)
e
-
(
∑
h
=
0
r
+
1
‍
ξ
h
t
h
); e(∑h=0r+1‍ξhth); ξr+1≠0
x
+
m
y
D
x
m
-
1
+
∑
h
=
1
r
+
1
h
ξ
h
D
x
h
-
1; Dx
e
(
∑
h
=
0
r
+
1
‍
ξ
h
t
h
+
x
t
+
y
t
m
)
=
∑
n
=
0
∞
H
A
n
(
m
)
(
x
,
y
)
t
n
n
!
The Gould-Hopper-generalized Gould-Hopper polynomials (forr=1, the Gould-Hopper-Hermite polynomials HHn(m)(x,y) and for r=2, the Gould-Hopper classical 2-orthogonal polynomials)
(XIII)
(
1
-
t
)
s
+
1; 1(1-t)s+1
x
+
m
y
D
x
m
-
1
-
m
+
1
1
-
D
x; Dx
1
(
1
-
t
)
s
+
1
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
G
n
(
m
,
s
)
(
x
,
y
)
t
n
The Gould-Hopper-Miller-Lee polynomials (fors=0, the Gould-Hopper-truncated exponential polynomials Henm(x,y) and for s=β-1 Gould-Hopper-modified Laguerre polynomials Hfn(m,β)(x,y))
(XIV)
(
e
t
+
1
)
2
t; 2t(et+1)
x
+
m
y
D
x
m
-
1
+
e
D
x
(
1
-
D
x
)
+
1
(
e
D
x
+
1
); Dx
2
t
(
e
t
+
1
)
e
x
t
+
y
t
m
=
∑
n
=
0
∞
H
G
n
(
m
)
(
x
,
y
)
t
n
n
!
The Gould-Hopper-Genocchi polynomialsNote 2.
Since, form=2, the GHAP HAnm(x,y) reduce to the HAP HAn(x,y) therefore, for m=2, Table 2 gives the list of the corresponding HAP HAn(x,y).The results established in this paper are general and include new families of special polynomials, consequently introducing many new special polynomials.



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*Source: 328032-2013-02-21.xml*