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Mathematical Sciences (2019) 13:153–164 https://doi.org/10.1007/s40096-019-0286-4
ORIGINAL RESEARCH
A linear algebra approach to the hybrid Sheffer–Appell polynomials
Subuhi Khan1 · Mahvish Ali1
Received: 5 September 2017 / Accepted: 17 April 2019 / Published online: 25 April 2019 © The Author(s) 2019
AbstractIn this article, a method based on linear algebra approach is adopted to study the hybrid Sheffer–Appell polynomials. The recursive formulas and differential equation for these polynomials are derived by using the properties and relationships between the Pascal functional matrices and the Wronskian matrices. The corresponding results for some mixed type special polynomials are also obtained.
Keywords Sheffer polynomials · Generalized Pascal functional matrix · Wronskian matrix · Recursive formulas · Differential equation
Mathematics Subject Classification 15A15 · 15A24 · 33C45 · 65QXX
Introduction and preliminaries
Recently, a systematic study of certain new classes of mixed special polynomials related to the Appell and Sheffer poly-nomial sequences is introduced and studied, see for example [3–7]. These mixed special polynomials are important due to the fact that they posses important properties such as dif-ferential equations, generating functions, series definitions, integral representations etc. The problems arising in differ-ent areas of science and engineering are usually expressed in terms of differential equations, which in most of the cases have special functions as their solutions. Recently, Srivas-tava et al. [12] derived the differential, integro-differential and partial differential equations for the Hermite-based Appell polynomials [7] using the factorization method. Further, the same method is extended by Khan and Riyasat [5] to derive a set of differential equations of finite order for the 2-iterated Appell polynomials [3]. In this article, recur-sive formulas and differential equation for the hybrid Shef-fer–Appell polynomials are derived by applying the algebra of Pascal and Wronskian matrices.
We review some definitions and concepts related to the Pascal and Wronskian matrices which will be used in Sects. 2 and 3.
Let = {h(t) =∑∞
k=0ak
tk
k!�ak ∈ ℂ} be the ℂ-algebra of
formal power series.For h(t) ∈ , the generalized Pascal functional matrix
[13] of an analytic function h(t) denoted by n[h(t)] is a square matrix of order (n + 1) defined as :
It should be noted that h(k) denotes the kth order derivative of h and hk denotes the kth power of h throughout the article.
Also, the nth order Wronskian matrix of analytic func-tions h1(t), h2(t), h3(t),… , hm(t) is an (n + 1) × m matrix and is defined as:
(1)
n[h(t)]i,j =
⎧⎪⎨⎪⎩
�i
j
�h(i−j)(t), if i ≥ j, i, j = 0, 1, 2,… , n
0, otherwise.
(2)
n[h1(t), h2(t), h3(t),… , hm(t)]
=
⎡⎢⎢⎢⎣
h1(t) h2(t) h3(t) ⋯ hm(t)
h�
1(t) h
�
2(t) h
�
3(t) ⋯ h
�
m(t)
⋮ ⋮ ⋮ ⋱ ⋮
h(n)
1(t) h
(n)
2(t) h
(n)
3(t) ⋯ h(n)
m(t)
⎤⎥⎥⎥⎦.
* Mahvish Ali [email protected]
Subuhi Khan [email protected]
1 Department of Mathematics, Aligarh Muslim University, Aligarh, India
154 Mathematical Sciences (2019) 13:153–164
1 3
It is important to note that in the expressions n[h(x, t)]t=0 and n[h(x, t)]t=0 , we consider t as the working variable and x as a parameter.
We recall certain important properties and relationships between the Pascal functional and Wronskian matrices [14].
For any a, b ∈ ℂ and any analytic functions h(t), l(t) ∈ , the following properties hold true:
where Λn = diag[0!, 1!, 2!,… , n!] and h(0) = 0 and h�(0) ≠ 0.Fur ther, for any analytic functions l( t) and
h1(t), h2(t),… , hm(t) , the following property holds true:
One of the important classes of polynomial sequences is the class of Appell polynomial sequences [1]. The Appell polynomials constitute an important class of polynomials because of their remarkable applications in numerous fields. The Appell polynomials appear in different applications in pure and applied mathematics. These polynomials arise in theoretical physics, chemistry and several branches of mathematics such as the study of polynomial expansions of analytic functions, number theory and numerical analysis.
In 1880, Appell [1] introduced and studied sequences of n-degree polynomials An(x), n = 0, 1, 2,… , satisfying the recurrence relation
The generating function of the sequence of polynomials An(x) is given as:
where
Another, important class of polynomial sequences is the class of Sheffer sequences. The Sheffer sequences [11] arise
(3)n[ah(t) + bl(t)] = an[h(t)] + bn[l(t)].
(4)n[ah(t) + bl(t)] = an[h(t)] + bn[l(t)].
(5)n[l(t)]n[h(t)] = n[h(t)]n[l(t)] = n[h(t)l(t)].
(6)n[l(t)]n[h(t)] = n[h(t)]n[l(t)] = n[h(t)l(t)].
(7)n[l(h(t))]t=0 = n[1, h(t), h
2(t), h3(t),
… , hn(t)]t=0Λ−1nn[l(t)]t=0,
(8)n[l(t)]n[h1(t), h2(t),… , hm(t)]
= n[(lh1)(t), (lh2)(t),… , (lhm)(t)].
(9)d
dxAn(x) = nAn−1(x), n = 1, 2,… .
(10)ext
a(t)=
∞∑n=0
An(x)tn
n!,
(11)a(t) =
∞∑n=0
ℵn
tn
n!, ℵ0 ≠ 0.
in numerous problems of applied mathematics, theoretical physics, approximation theory and several other mathemati-cal branches. According to Roman [10], the Sheffer sequence sn(x) is uniquely determined by two (formal) power series:
and
Then, the exponential generating function of sn(x) is given by:
for all x in ℂ , where f −1(t) is the compositional inverse of f(t).
It should be noted that for �(t) = 1 , the Sheffer sequence sn(x) becomes the associated Sheffer sequence �n(x) and for f (t) = t , it becomes the Appell sequence An(x).
Certain members belonging to the Sheffer associated Sheffer and Appell families are given in the Table 1.
By combining the Sheffer and Appell sequences, an extended class of Sheffer polynomials namely the Shef-fer–Appell polynomials denoted by sAn(x) is introduced and studied in [4], this class is defined by the following generat-ing function:
where �(t), a(t) are the inver t ible ser ies with �(0) ≠ 0, a(0) ≠ 0 and f(t) is a delta series with f (0) = 0 and f �(0) ≠ 0.
We remark that since the l.h.s. of definition (15) includes �(t) , f(t) of Sheffer class, while a(t) in the denominator is corresponding to the Appell class. Therefore, it is justified to call this hybrid family as Sheffer–Appell family. The main advantage of this family is that it allows to consider mixed type special polynomials by taking �(t) , f(t) of the members belonging to the Sheffer family and a(t) of the members belonging to the Appell family.
In [14], Youn and Yang derived the differential equa-tion and recursive formulas for the Sheffer polynomial sequences using the generalized Pascal functional matrix of an analytic function and Wronskian matrix of several ana-lytic functions. In this article, the method adopted in [14] is extended to derive the recursive formulas and differential
(12)f (t) =
∞∑n=0
fntn
n!, f0 = 0, f1 ≠ 0
(13)�(t) =
∞∑n=0
�ntn
n!, �0 ≠ 0.
(14)exf
−1(t)
�(f −1(t))=
∞∑n=0
sn(x)tn
n!,
(15)exf
−1(t)
�(f −1(t))a(t)=
∞∑n=0
sAn(x)tn
n!,
155Mathematical Sciences (2019) 13:153–164
1 3
equation satisfied by the Sheffer–Appell polynomials sAn(x) . In Sect. 2, some recursive formulas satisfied by sAn(x) are established. In Sect. 3, differential equation for the Shef-fer–Appell polynomials is derived. Certain examples are considered in Sect. 4.
Recursive formulas
In order to utilize the Wronskian matrices, the vector form of the Sheffer–Appell polynomial sequence is required.
The Sheffer–Appell vector denoted by sn(x) is defined as:
where {sAn(x)} is the Sheffer–Appell polynomials sequence defined by Eq. (15).
Since exf−1(t)
�(f−1(t))a(t) is analytic, therefore by Taylor’s theorem,
it follows that
In view of Eq. (17), the Sheffer–Appell vector (16) can be expressed as:
First, we prove the following Lemma:
(16)sn(x) = [sA0(x) sA1(x) sA2(x) … sAn(x)]T,
(17)sAk(x) =(d
dt
)k exf−1(t)
�(f −1(t))a(t)
|||||t=0, k ≥ 0.
(18)sn(x) =[sA0(x) sA1(x) sA2(x) … sAn(x)]
T
=n
[exf
−1(t)
�(f −1(t))a(t)
]
t=0
.
Lemma 2.1 Let {sAn(x)} be the Sheffer–Appell polynomial sequence. Then,
Proof By making use of property (6) in the r.h.s. of Eq. (18) and then using property (7) in the second term on r.h.s. of the resultant equation, it follows that
In view of the fact that n[ext]t=0 = [1 x x2 … xn]T and
using property (6), Eq. (20) takes the form
(19)
(n[sA0(x), sA1(x), sA2(x), … , sAn(x)]
)TΛ−1
n
= n
[1
a(t)
]
t=0
n
[1, f −1(t), (f −1(t))2,
… , (f −1(t))n]t=0
× Λ−1nn
[1
�(t)
]
t=0
n[ext]t=0.
(20)
[sA0(x) sA1(x) sA2(x) … sAn(x)]T
= n
[1
a(t)
]
t=0
n[1, f−1(t), (f −1(t))2,… , (f −1(t))n]t=0
× Λ−1nn
[ext
�(t)
]
t=0
.
(21)
[sA0(x) sA1(x) sA2(x) … sAn(x)]T
= n
[1
a(t)
]
t=0
n[1, f−1(t), (f −1(t))2,… , (f −1(t))n]t=0
× Λ−1nn
[1
�(t)
]
t=0
[1 x x2 … xn]T.
Table 1 Certain members of the Sheffer, associated Sheffer and Appell families
Sheffer polynomials
S. no. �(t);f (t);f −1(t) Generating function Polynomials
I.e
t2
4 ;t
2;2t e2xt−t
2
=∑∞
n=0Hn(x)
tn
n!Hermite polynomials Hn(x) [9]
II. (1 − t)−1;t
t−1;
t
t−11
(1−t)exp
�xt
t−1
�=∑∞
n=0Ln(x)t
nLaguerre polynomials n!Ln(x) [9]
Associated Sheffer polynomials
S. no. f(t); f −1(t) Generating function Polynomials
I. ln(1 + t) ; et − 1 exp(x(et − 1)) =∑∞
n=0�n(x)
tn
n!Exponential polynomials�n(x) [2]
II. −1
2t2 + t ; 1 −
√1 − 2t exp
�x(1 −
√1 − 2t)
�=∑∞
n=0pn(x)
tn
n!
Bessel polynomials pn(x) [8]
Appell polynomials
S. no. a(t) Generating function Polynomials
I. et−1
t
t
(et−1)ext =
∑∞
n=0Bn(x)
tn
n!Bernoulli polynomials Bn(x) [9]
II. et+1
2
2
(et+1)ext =
∑∞
n=0En(x)
tn
n!Euler polynomials En(x) [9]
156 Mathematical Sciences (2019) 13:153–164
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Differentiation of Eq. (21) k times with respect to x and division by k! gives
The l.h.s. of Eq. (22) is the kth column of
and the r.h.s. of Eq. (22) is the kth column of
Hence, assertion (19) follows.Now, we establish certain recursive formulas for the Shef-
fer–Appell polynomials sAn(x) . ◻
First, we derive a recursive formula, which expresses sAn+1(x) in terms of sAn(x) and its derivatives by proving the following result:
Theorem 2.1 Let sAn(x) denote the Sheffer–Appell polyno-mials. Then sA0(x) =
1
�(0)a(0) and
where
Proof In view of definition (2) and Eq. (17), we have
(22)
1
k!
[sA
(k)
0(x) sA
(k)
1(x) … sA
(k)n(x)
]T
= n
[1
a(t)
]
t=0
n
[1, f −1(t), (f −1(t))2,… , (f −1(t))n
]t=0
Λ−1n
× n
[1
�(t)
]
t=0
[0⋯ 0 1
(k + 1
k
)x
(k + 2
k
)x2 …
(n
k
)xn−k
]T.
(n
[sA0(x) sA1(x) sA2(x) … sA
(k)n(x)
])TΛ−1
n
n
[1
a(t)
]
t=0
n[1, f−1(t), (f −1(t))2,
… , (f −1(t))n]t=0Λ−1nn
[1
�(t)
]
t=0
n[ext]t=0.
(23)sAn+1(x) =
n∑k=0
(�kx + �k + �k)sA
(k)n(x)
k!, n ≥ 0,
�k =
(1
f �(t)
)(k)|||t=0; �k =(−
��(t)
�(t)f �(t)
)(k)|||t=0; �k
=
(−a�(f (t))
a(f (t))
)(k)|||t=0.
(24)n
[d
dt
(exf
−1(t)
�(f −1(t))a(t)
)]
t=0
= [sA1(x) sA2(x) sA3(x) … sAn+1(x)]T.
Per forming the di f ferent ia t ion in express ion n
[d
dt
(exf
−1 (t)
�(f−1(t))a(t)
)]t=0
and using properties (5)–(7) in a suit-able manner, we have
Further, in view of Lemma 2.1, we have
Equating the last rows of Eqs. (24) and (26), we get assertion (23). ◻
Remark 2.1 Since �(t) = 1 ⟹ �k = 0 (k ≥ 0) , therefore for �(t) = 1 , we deduce the following consequence of Theorem 2.1:
Corollary 2.1 Let �An(x) denote the associated Sheffer–
Appell polynomials. Then �A0(x) =
1
a(0) and
(25)
n
[d
dt
(exf
−1(t)
�(f −1(t))a(t)
)]
t=0
= n
[(x −
��(f −1(t))
�(f −1(t))
)exf
−1(t)
a(t)f �(f −1(t))�(f −1(t))
−a�(t)exf
−1(t)
(a(t))2�(f −1(t))
]
t=0
= n
[1
a(t)
]n[1, f
−1(t), (f −1(t))2,… , (f −1(t))n]t=0Λ−1n
×n
[(x −
��(t)
�(t)
)ext
f �(t)�(t)−
a�(f (t))ext
a(f (t))�(t)
]
t=0
= n
[1
a(t)
]
t=0
n[1, f−1(t), (f −1(t))2,… , (f −1(t))n]t=0Λ
−1n
× n
[1
�(t)
]
t=0
n[ext]t=0n
[x
f �(t)−
��(t)
�(t)f �(t)−
a�(f (t))
a(f (t))
]
t=0
.
(26)
n
�d
dt
�exf
−1(t)
�(f −1(t))a(t)
��
t=0
=�n[sA0(x), sA1(x), sA2(x), … , sAn(x)]
�TΛ−1
n
×n
�x
f �(t)−
��(t)
�(t)f �(t)−
a�(f (t))
a(f (t))
�
t=0
=
⎡⎢⎢⎢⎢⎢⎢⎣
sA0(x) 0 0 ⋯ 0
sA1(x)sA
�1(x)
1!0 ⋯ 0
sA2(x)sA
�2(x)
1!
sA��2(x)
2!⋯ 0
⋮ ⋮ ⋮ ⋱ ⋮
sAn(x)sA
�n(x)
1!
sA��n(x)
2!⋯ sA
(n)n (x)
n!
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎣
�0x + �0 + �0�1x + �1 + �1
⋮
�nx + �n + �n
⎤⎥⎥⎥⎦.
(27)�An+1(x) =
n∑k=0
(�kx + �k)�A(k)n(x)
k!, n ≥ 0,
157Mathematical Sciences (2019) 13:153–164
1 3
where
Remark 2.2 Since f (t) = t ⟹ �0 = 1 ; �k = 0 (k ≠ 0) , there-fore for f (t) = t , we deduce the following consequence of Theorem 2.1:
Corollary 2.2 Let A[2]n(x) denote the 2-iterated Appell poly-
nomials. Then A[2]
0(x) =
1
�(0)a(0) and
where
Next, we derive a pure recurrence relation which expresses sAn+1(x) in terms of sAk(x) ( k = 0, 1,… , n ) in the form of following theorem:
Theorem 2.2 Let sAn(x) denotes the Sheffer–Appell poly-nomials. Then sA0(x) =
1
�(0)a(0) and
where
Proof Using property (6) in expression n
[f �(f −1(t))
d
dt
(ex(f
−1 (t))
�(f−1(t))a(t)
)]t=0
, it follows that
�k =
(1
f �(t)
)(k)|||t=0; �k =(−a�(f (t))
a(f (t))
)(k)|||t=0.
(28)
A[2]
n+1(x) = xA[2]
n(x) +
n∑k=0
(�k + �k)(A[2]
n(x))(k)(x)
k!, n ≥ 0,
�k =
(−��(t)
�(t)
)(k)|||t=0; �k =(−a�(t)
a(t)
)(k)|||t=0.
(29)
�0 sAn+1(x) =x sAn(x) +
n∑k=0
(n
k
)(�k + �k)sAn−k(x)
−
n∑k=1
(n
k
)�k sAn+1−k(x), n ≥ 0,
�k =(f�(f −1(t)))(k)
|||t=0 =(
1
(f −1(t))�
)(k)|||t=0;
�k =
(−��(f −1(t))
�(f −1(t))
)(k)|||t=0;
�k =
(−f �(f −1(t))
a�(t)
a(t)
)(k)|||t=0.
On the other hand, performing the differentiation in the same expression and using properties (4) and (6), it follows that
◻
(30)
n
�f �(f −1(t))
d
dt
�ex(f
−1(t))
�(f −1(t))a(t)
��
t=0
= n
�d
dt
�ex(f
−1(t))
�(f −1(t))a(t)
��
t=0
n[f�(f −1(t))]t=0
=
⎡⎢⎢⎢⎢⎢⎢⎢⎣
sA1(x) 0 ⋯ 0
sA2(x) sA1(x) ⋯ 0
sA3(x)
�2
1
�sA2(x) ⋯ 0
⋮ ⋮ ⋱ ⋮
sAn+1(x)
�n
1
�sAn(x) ⋯ sA1(x)
⎤⎥⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎣
�0�1�2⋮
�n
⎤⎥⎥⎥⎥⎥⎦
.
(31)
n
�f �(f −1(t))
d
dt
�ex(f
−1(t))
�(f −1(t))a(t)
��
t=0
= n
�xexf
−1(t)
�(f −1(t))a(t)−
��(f −1(t))exf
−1(t)
�(f −1(t))2a(t)
−f �(f −1(t))a�(t)exf
−1(t)
a(t)2�(f −1(t))
�
t=0
= xn
�ex(f
−1(t))
�(f −1(t))a(t)
�
t=0
+ n
�ex(f
−1(t))
�(f −1(t))a(t)
�
t=0
×n
�−��(f −1(t))
�(f −1(t))− f �(f −1(t))
a�(t)
a(t)
�
t=0
= x
⎡⎢⎢⎢⎢⎢⎣
sA0(x)
sA1(x)
sA2(x)
⋮
sAn(x)
⎤⎥⎥⎥⎥⎥⎦
+
⎡⎢⎢⎢⎢⎢⎢⎢⎣
sA0(x) 0 ⋯ 0
sA1(x) sA0(x) ⋯ 0
sA2(x)
�2
1
�sA1(x) ⋯ 0
⋮ ⋮ ⋱ ⋮
sAn(x)
�n
1
�sAn−1(x) ⋯ sA0(x)
⎤⎥⎥⎥⎥⎥⎥⎥⎦
×
⎡⎢⎢⎢⎢⎢⎣
�0 + �0�1 + �1�2 + �2
⋮
�n + �n
⎤⎥⎥⎥⎥⎥⎦
.
158 Mathematical Sciences (2019) 13:153–164
1 3
Equating the last rows of Eqs. (30) and (31), we get asser-tion (29).
Remark 2.3 Since �(t) = 1 ⟹ �k = 0 (k ≥ 0) , therefore for �(t) = 1 , we deduce the following consequence of Theorem 2.2:
Corollary 2.3 Let �An(x) denote the associated Sheffer–
Appell polynomials. Then �A0(x) =
1
a(0) and
where
Remark 2.4 Since f (t) = t ⟹ �0 = 1 ; �k = 0 (k ≠ 0) , there-fore for f (t) = t , we deduce the following consequence of Theorem 2.2:
Corollary 2.4 Let A[2]n(x) denote the 2-iterated Appell poly-
nomials. Then A[2]
0(x) =
1
�(0)a(0) and
(32)
�0 �An+1(x) = x
�An(x)
+
n∑k=0
(n
k
)�k �
An−k(x) −
n∑k=1
(n
k
)�k �
An+1−k(x), n ≥ 0,
�k = (f �(f −1(t)))(k)|||t=0 =
(1
(f −1(t))�
)(k)|||t=0; �k
=
(−f �(f −1(t))
a�(t)
a(t)
)(k)|||t=0.
where
Finally, we derive a pure recurrence relation, which provides a representation of sAn+1(x) in terms of sAk(x) ( k = 0, 1, 2,… n ) by proving the following result:
Theorem 2.3 Let sAn(x) denote the Sheffer–Appell polyno-mials. Then sA0(x) =
1
�(0)a(0) and
where
Proof Performing the differentiation in expression n
[d
dt
(exf
−1 (t)
�(f−1(t))a(t)
)]t=0
and then using property (6), we have
(33)
A[2]
n+1(x) = x A[2]
n(x) +
n∑k=0
(n
k
)(�k + �k)A
[2]
n−k(x), n ≥ 0,
�k =
(−��(t)
�(t)
)(k)|||t=0; �k =(−a�(t)
a(t)
)(k)|||t=0.
(34)sAn+1(x) =
n∑k=0
(n
k
)(x�k + �k + �k)sAn−k(x), n ≥ 0,
�k =
(1
f �(f −1(t))
)(k)|||t=0; �k
=
(−��(f −1(t))
�(f −1(t))
1
f �(f −1(t))
)(k)|||t=0; �k =(−a�(t)
a(t)
)(k)|||t=0.
(35)
n
�d
dt
�exf
−1(t)
�(f −1(t))a(t)
��
t=0
= n
��x −
��(f −1(t))
�(f −1(t))
�1
f �(f −1(t))−
a�(t)
a(t)
�n
�exf
−1(t)
�(f −1(t))a(t)
�
t=0
=
⎡⎢⎢⎢⎢⎢⎢⎢⎣
x�0 + �0 + �0 0 0 ⋯ 0
x�1 + �1 + �1 x�0 + �0 + �0 0 ⋯ 0
x�2 + �2 + �2
�2
1
�x�1 + �1 + �1 x�0 + �0 + �0 ⋯ 0
⋮ ⋮ ⋮ ⋱ ⋮
x�n + �n + �n
�n
1
�x�n−1 + �n−1 + �n−1
�n
2
�x�n−2 + �n−2 + �n−2 ⋯ x�0 + �0 + �0
⎤⎥⎥⎥⎥⎥⎥⎥⎦
×
⎡⎢⎢⎢⎢⎢⎣
sA0(x)
sA1(x)
sA2(x)
⋮
sAn(x)
⎤⎥⎥⎥⎥⎥⎦
.
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1 3
Equating the last rows of Eqs. (24) and (35), we get assertion (34). ◻
Remark 2.5 Since �(t) = 1 ⟹ �k = 0 (k ≥ 0) , there-fore for �(t) = 1 , we deduce the following consequence of Theorem 2.3:
Corollary 2.5 Let �An(x) denote the associated Sheffer–
Appell polynomials. Then �A0(x) =
1
a(0) and
where
Remark 2.6 We note that f (t) = t ⟹ �0 = 1 ; �k = 0 (k ≠ 0) . Therefore, taking f (t) = t , in Theorem 2.3, we get Corollary 2.4.
In the next section, differential equation for the Shef-fer–Appell polynomials sAn(x) is derived.
Differential equation
In order to derive the differential equation for the Shef-fer–Appell polynomials sAn(x) , we prove the following result:
Theorem 3.1 The Sheffer–Appell polynomials sAn(x) satisfy the following differential equation:
where
(36)�An+1(x) =
n∑k=0
(n
k
)(x�k + �k)�An−k(x), n ≥ 0,
�k =
(1
f �(f −1(t))
)(k)|||t=0; �k =(−a�(t)
a(t)
)(k)|||t=0.
(37)n∑
k=1
(�kx + �k + �k)sA
(k)n(x)
k!− n sAn(x) = 0,
�k =
(−��(t)f (t)
�(t)f �(t)
)(k)|||t=0; �k =(
f (t)
f �(t)
)(k)|||t=0;
�k =
(−f (t)a�(f (t))
a(f (t))
)(k)|||t=0.
Proof In view of proper ty (6), the expression n
[td
dt
(exf
−1 (t)
�(f−1(t))a(t)
)]t=0
can be written as:
On the other hand performing the differentiation in the same expression and using properties (5)–(7) in a suitable man-ner, we have
Again, in view of Lemma 2.1, we have
(38)
n
�td
dt
�exf
−1(t)
�(f −1(t))a(t)
��
t=0
= n[t]t=0n
�d
dt
�exf
−1(t)
�(f −1(t))a(t)
��
t=0
=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
0 0 0 0 ⋯ 0 0 0
1 0 0 0 ⋯ 0 0 0
0 2 0 0 ⋯ 0 0 0
0 0 3 0 ⋯ 0 0 0
⋮ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋮
⋱ ⋱
0 0 0 0 ⋯ n − 1 0 0
0 0 0 0 ⋯ 0 n 0
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎣
sA1(x)
sA2(x)
sA3(x)
⋮
sAn(x)
sAn+1(x)
⎤⎥⎥⎥⎥⎥⎥⎦
.
(39)
n
[td
dt
(exf
−1(t)
�(f −1(t))a(t)
)]
t=0
= n
[(x −
��(f −1(t))
�(f −1(t))
)texf
−1(t)
a(t)f �(f −1(t))�(f −1(t))
−ta�(t)exf
−1(t)
(a(t))2�(f −1(t))
]
t=0
= n
[1
a(t)
]n[1, f
−1(t), (f −1(t))2,… , (f −1(t))n]t=0Λ−1n
×n
[(x −
��(t)
�(t)
)f (t)ext
f �(t)�(t)−
f (t)a�(f (t))ext
a(f (t))�(t)
]
t=0
= n
[1
a(t)
]
t=0
n[1, f−1(t), (f −1(t))2,… , (f −1(t))n]t=0Λ
−1n
× n
[1
�(t)
]
t=0
n[ext]t=0n
[xf (t)
f �(t)−
��(t)f (t)
�(t)f �(t)−
f (t)a�(f (t))
a(f (t))
]
t=0
.
160 Mathematical Sciences (2019) 13:153–164
1 3
On equating the last rows of Eqs. (38) and (40) and using the fact that
we get assertion (37). ◻
Remark 3.1 For �(t) = 1 , the Sheffer–Appell polynomials sAn(x) reduce to the associated Sheffer–Appell polynomi-als
�An(x) and since �(t) = 1 ⟹ �k = 0 (k ≥ 1) , there-
fore, for �(t) = 1 , we deduce the following consequence of Theorem 3.1:
Corollary 3.1 The associated Sheffer–Appell polynomials �An(x) satisfy the following differential equation:
where
Remark 3.2 For f (t) = t , the Sheffer–Appell polynomials sAn(x) reduce to the 2-iterated Appell polynomials A[2]
n(x)
[3] and since f (t) = t ⟹ �1 = 1 ; �k = 0 (k ≠ 1) , there-fore, for f (t) = t , we deduce the following consequence of Theorem 3.1:
Corollary 3.2 The 2-iterated Appell polynomials A[2]n(x)
satisfy the following differential equation:
where
(40)
n
�td
dt
�exf
−1(t)
�(f −1(t))a(t)
��
t=0
=�n[sA0(x), sA1(x), sA2(x), … , sAn(x)]
�TΛ−1
n
×n
�xf (t)
f �(t)−
��(t)f (t)
�(t)f �(t)−
f (t)a�(f (t))
a(f (t))
�
t=0
=
⎡⎢⎢⎢⎢⎢⎢⎣
sA0(x) 0 0 ⋯ 0
sA1(x)sA
�1(x)
1!0 ⋯ 0
sA2(x)sA
�2(x)
1!
sA��2(x)
2!⋯ 0
⋮ ⋮ ⋮ ⋱ ⋮
sAn(x)sA
�n(x)
1!
sA��n(x)
2!⋯ sA
(n)n (x)
n!
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎣
�0x + �0 + �0�1x + �1 + �1
⋮
�nx + �n + �n
⎤⎥⎥⎥⎦.
f (0) = 0 ⟹ �0 = �0 = �0 = 0,
(41)n∑
k=1
(�kx + �k)�A(k)n(x)
k!− n
�An(x) = 0,
�k =
(f (t)
f �(t)
)(k)|||t=0; �k =(−f (t)a�(f (t))
a(f (t))
)(k)|||t=0.
(42)x(A[2]n(x))� +
n∑k=1
(�k + �k)(A[2]
n(x))(k)
k!− nA[2]
n(x) = 0,
In the next section, we obtain the differential equations and recursive formulas for some members belonging to the Sheffer–Appell, associated Sheffer–Appell and 2-iterated Appell families.
Examples
We derive the differential equations and recursive formu-las for some members belonging to the Sheffer–Appell family by applying Theorem 3.1 and Theorems 2.1–2.3, respectively.
Example 4.1 For �(t) = et2
4 and f (t) = t
2 , the Sheffer poly-
nomials become the Hermite polynomials Hn(x) and for a(t) =
et−1
t , the Appell polynomials become the Bernoulli
polynomials Bn(x) . Therefore, for these values of �(t) , f(t) and a(t), the Sheffer–Appell polynomials reduce to the Her-mite-Bernoulli polynomials HBn(x).
From Theorem 3.1, we find
Substituting the values from Eq. (43) into Eq. (37), we obtain the following differential equation for the Hermite-Bernoulli polynomials HBn(x):
�k =
(−t��(t)
�(t)
)(k)|||t=0; �k =
(−ta�(t)
a(t)
)(k)|||t=0.
(43)
�k =
�−t2
2
�(k)���t=0; �k = (t)(k)��t=0; �k =
⎛⎜⎜⎝−
t
2e
t
2 − et
2 + 1
et
2 − 1
⎞⎟⎟⎠
(k)
���t=0.
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1 3
Similarly, using Theorems 2.1, 2.2 and 2.3, the following recursive formulas for the Hermite-Bernoulli polynomials HBn(x) are obtained:
and
Example 4.2 For �(t) = 1
1−t and f (t) = t
t−1 , the Sheffer
polynomials become the Laguerre polynomials Ln(x) and for a(t) = et+1
2 , the Appell polynomials become the Euler
polynomials En(x) . Therefore, for these values of �(t) , f(t) and a(t), the Sheffer–Appell polynomials reduce to the Laguerre–Euler polynomials LEn(x).
From Theorem 3.1, we find
Substituting the values from Eq. (48) into Eq. (37), we obtain the following differential equation for the Laguerre–Euler polynomials LEn(x):
(44)HB
��n(x) − 2xHB
�n(x) − 2
n�k=1
⎛⎜⎜⎝−
t
2e
t
2 − et
2 + 1
et
2 − 1
⎞⎟⎟⎠
(k)�����t=0HB
(k)n(x)
k!+ 2nHBn(x) = 0.
(45)HB0(x) = 1,
(46)
HBn+1(x) = 2xHBn(x) −H B�n(x)
+
n�k=0
⎛⎜⎜⎝−
t
2e
t
2 − et
2 + 1
t
2(e
t
2 − 1)
⎞⎟⎟⎠
(k)�����t=0HB
(k)n(x)
k!, n ≥ 0
(47)
HBn+1(x) = 2xHBn(x) − 2nHBn−1(x)
+
n∑k=0
(n
k
)(−tet − et + 1
t(et − 1)
)(k)|||||t=0HBn−k(x), n ≥ 0.
(48)
�k =(−t)(k)��t=0; �k = (−t(t − 1))(k)��t=0;
�k =
⎛⎜⎜⎜⎝−
tet
t−1
(t − 1)�e
t
t−1 + 1
�⎞⎟⎟⎟⎠
(k)
�����t=0.
(49)
xLE��n(x) − (x − 1)LE
�n(x) −
n�k=1
⎛⎜⎜⎜⎝−
tet
t−1
(t − 1)�e
t
t−1 + 1
�⎞⎟⎟⎟⎠
(k)
�����t=0LE
(k)n(x)
k!+ nLEn(x) = 0.
Similarly, using Theorems 2.1, 2.2 and 2.3, the following recursive formulas for the Laguerre–Euler polynomials LEn(x) are obtained:
and
Next, we apply Corollary 3.1 to derive the differential equa-tions and Corollaries 2.1, 2.3 and 2.5 to derive the recursive formulas for some members belonging to the associated Sheffer–Appell family.
Example 4.3 For f (t) = ln(1 + t) , the associated Sheffer polynomials become the exponential polynomials �n(x) and for a(t) = et−1
t , the Appell polynomials become the Bernoulli
polynomials Bn(x) . Therefore, for these values of f(t) and a(t), the associated Sheffer–Appell polynomials reduce to the exponential-Bernoulli polynomials �Bn(x).
From Corollary 3.1, we find
(50)LE0(x) = 1,
(51)
LEn+1(x) = (1 − x)LEn(x) + (2x − 1)LE�n(x) − xLE
��n(x)
+
n∑k=0
(−
et
t−1
et
t−1 + 1
)(k)|||||t=0LE
(k)n(x)
k!,
n ≥ 0,
(52)
LEn+1(x) = (1 − x + 2n)LEn(x) − n2 LEn−1(x)
−
n∑n=0
(n
k
)((t − 1)2et
et + 1
)(k)|||||t=0LEn−k(x),
n ≥ 0
(53)
LEn+1(x) =
n∑k=0
n!
(n − k)!
((−1)k+1(k + 1)x + 1
)LEn−k(x)
+
(n
k
)(−
et
et + 1
)(k)|||||t=0LEn−k(x),
n ≥ 0.
(54)
�k = ((1 + t) ln(1 + t))(k)||t=0; �k =
(t − (1 + t) ln(1 + t)
t
)(k)|||t=0.
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1 3
From Eqs. (54) and (41), the following differential equa-tion for the exponential-Bernoulli polynomials �Bn(x) is obtained:
Similarly, using Corollaries 2.1, 2.3 and 2.5, the following recursive formulas for the exponential-Bernoulli polynomi-als �Bn(x) are obtained:
and
Example 4.4 For f (t) = −1
2t2 + t , the associated Sheffer
polynomials become the Bessel polynomials pn(x) and for a(t) =
et+1
2 , the Appell polynomials become the Euler poly-
nomials En(x) . Therefore, for these values of f(t) and a(t), the associated Sheffer–Appell polynomials reduce to the Bes-sel–Euler polynomials pEn(x).
From Corollary 3.1, we find
(55)
x�B�n(x) + x
�B��n(x)
2!+
n∑k=3
(−1)(k−2)x�B
(k)n(x)
k(k − 1)
+
n∑k=1
(t − (1 + t) ln(1 + t)
t
)(k)|||||t=0�B
(k)n(x)
k!
− n �Bn(x) = 0.
(56)�B0(x) = 1,
(57)
�Bn+1(x) = x(�Bn(x) +� B�n(x))
+
n∑k=0
(t − (1 + t) ln(1 + t)
t ln(1 + t)
)(k)|||||t=0�B
(k)n(x)
k!, n ≥ 0,
(58)
�Bn+1(x) = x�Bn(x) −
n∑k=1
(n
k
)(−1)k
�Bn+1−k(x)
+
n∑k=0
(n
k
)(et − tet − 1
tet(et − 1)
)(k)|||||t=0�Bn−k(x),
n ≥ 0
(59)�Bn+1(x) =
n∑k=0
(n
k
)(x +
(et − tet − 1
t(et − 1)
)(k)|||||t=0
)
�Bn−k(x), n ≥ 0.
(60)
�k =
�t(1 −
t
2)
1 − t
�(k)
���t=0; �k =
⎛⎜⎜⎝−t(1 −
t
2)et(1−
t
2)
et(1−
t
2) + 1
⎞⎟⎟⎠
(k)
���t=0.
From Eqs. (60) and (41), the following differential equation for the Bessel–Euler polynomials pEn(x) is obtained:
Similarly, using Corollaries 2.1, 2.3 and 2.5, the following recursive formulas for the Bessel–Euler polynomials pEn(x) are obtained:
and
Further, we apply Corollary 3.2 to derive the differ-ential equations and Corollaries 2.2 and 2.4 to derive the recursive formulas for some members belonging to the 2-iterated Appell family.
Example 4.5 For �(t) = a(t) =et−1
t , the 2-iterated Appell
polynomials reduce to the 2-iterated Bernoulli polynomials B[2]n(x).
From Corollary 3.2, we find
(61)
n�k=1
⎛⎜⎜⎝x
�t(1 −
t
2)
1 − t
�(k)�����t=0
+
⎛⎜⎜⎝−t(1 −
t
2)et(1−
t
2)
et(1−
t
2) + 1
⎞⎟⎟⎠
(k)�����t=0
⎞⎟⎟⎟⎠
pE(k)n(x)
k!− n pEn(x).
(62)pE0(x) = 1,
(63)
pEn+1(x) =
n�k=0
⎛⎜⎜⎝k!x +
�−
et(1−
t
2)
et(1−
t
2) + 1
�(k)�����t=0⎞⎟⎟⎠pE
(k)n(x)
k!, n ≥ 0,
(64)
pEn+1(x) = x pEn(x) −
n�k=1
�n
k
�(√1 − 2t)(k)
���t=0pEn+1−k(x)
+
n�k=0
�n
k
��−√1 − 2t
et
et + 1
�(k)�����t=0pEn−k(x), n ≥ 0
(65)
pEn+1(x) =
n∑k=0
(n
k
)(x((1 − 2t)1∕2)(k)
|||t=0
+
(−
et
et + 1
)(k)|||||t=0
)pEn−k(x), n ≥ 0.
(66)�k = �k =
(et − tet − 1
et − 1
)(k)|||t=0.
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1 3
Substituting the values from Eq. (66) in Eq. (42), we find the following differential equation for the 2-iterated Bernoulli polynomials B[2]
n(x):
Similarly, using Corollaries 2.2 and 2.4, the following recur-sive formulas for the 2-iterated Bernoulli polynomials B[2]
n(x)
are obtained:
and
Example 4.6 For �(t) = a(t) =et+1
2 , the 2-iterated Appell pol-
ynomials reduce to the 2-iterated Euler polynomials E[2]n(x).
From Corollary 3.2, we find
Substituting the values from Eq. (71) into Eq. (42), we find the following differential equation for the 2-iterated Euler polynomials E[2]
n(x):
Similarly, using Corollaries 2.2 and 2.4, the following recur-sive formulas for the 2-iterated Euler polynomials E[2]
n(x)
are obtained:
(67)x(B[2]
n(x))� + 2
n∑k=1
(et − tet − 1
et − 1
)(k)|||||t=0(B[2]
n(x))(k)
k!− n B[2]
n(x) = 0.
(68)B[2]
0(x) = 1,
(69)
B[2]
n+1(x) = xB[2]
n(x)
+ 2
n∑k=0
(n
k
)(et − tet − 1
t(et − 1)
)(k)|||||t=0(B[2]
n(x))(k)
k!, n ≥ 0
(70)
B[2]
n+1(x) =xB[2]
n(x) + 2
n∑k=0
(n
k
)
(et − tet − 1
t(et − 1)
)(k)|||||t=0B[2]
n−k(x), n ≥ 0.
(71)�k = �k =
(−
tet
et + 1
)(k)|||t=0.
(72)
x(E[2]n(x))� + 2
n∑k=1
(−
tet
et + 1
)(k)|||||t=0(E[2]
n(x))(k)
k!− n E[2]
n(x) = 0.
and
Example 4.7 For �(t) = et−1
t and a(t) = et+1
2 , the 2-iterated
Appell polynomials reduce to the Bernoulli–Euler polyno-mials BEn(x).
From Corollary 3.2, we find
Substituting the values from Eq. (76) into Eq. (42), the fol-lowing differential equation for the Bernoulli–Euler polyno-mials BEn(x) is obtained:
Similarly, using Corollaries 2.2 and 2.4, the following recursive formulas for the Bernoulli–Euler polynomials BEn(x) are obtained:
and
(73)E[2]
0(x) = 1,
(74)
E[2]
n+1(x) = xE[2]
n(x)
+ 2
n∑k=0
(n
k
)(−
et
et + 1
)(k)|||||t=0(E[2]
n(x))(k)
k!, n ≥ 0
(75)
E[2]
n+1(x) = xE[2]
n(x)
+ 2
n∑k=0
(n
k
)(−
et
et + 1
)(k)|||||t=0E[2]
n−k(x), n ≥ 0.
(76)�k =
(et − tet − 1
et − 1
)(k)|||t=0; �k =
(−
tet
et + 1
)(k)|||t=0.
(77)
x BE�n(x) +
n∑k=1
((et − tet − 1
et − 1
)(k)|||||t=0+
(−
tet
et + 1
)(k)|||||t=0
)BE
(k)n(x)
k!− n BEn(x) = 0.
(78)BE0(x) = 1,
(79)BEn+1(x) +
n∑k=0
((et − tet − 1
t(et − 1)
)(k)|||||t=0+
(−
tet
t(et + 1)
)(k)|||||t=0
)BE
(k)n(x)
k!, n ≥ 0
164 Mathematical Sciences (2019) 13:153–164
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It is to be noted that the differential equations and recur-sive formulas for other members belonging to the Shef-fer–Appell, associated Sheffer–Appell and 2-iterated Appell families can also be obtained in a similar manner by making suitable substitutions.
In this article, the differential equation and recursive formulas for the Sheffer–Appell polynomial sequences are established by using the Pascal and Wronskian matrices. The corresponding results are also established for the associated Sheffer–Appell and 2-iterated Appell polynomial sequences, which are the subclasses of the Sheffer–Appell polynomial sequences. Since the Sheffer–Appell polynomials are impor-tant from the point of view of their applications in various fields, therefore, the differential equation and recursive for-mulas satisfied by these polynomials may be used to solve the existing as well as new emerging problems in certain branches of science. This approach can be extended to derive the properties of other generalized hybrid special polynomial families.
Acknowledgements This work has been done under Senior Research Fellowship (Award Letter No. F1-17.1/2014-15/MANF-2014-15-MUS-UTT-34170/(SA-III/Website)) awarded to the second author by the University Grants Commission, Government of India, New Delhi.
Open Access This article is distributed under the terms of the Crea-tive Commons Attribution 4.0 International License (http://creat iveco mmons .org/licen ses/by/4.0/), which permits unrestricted use, distribu-tion, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
(80)
BEn+1(x) =x BEn(x) +
n∑k=0
(n
k
)((et − tet − 1
t(et − 1)
)(k)|||||t=0+
(−
tet
t(et + 1)
)(k)|||||t=0
)BEn−k(x),
n ≥ 0.
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