Certain generating funtion of generalized Apostol type Legendre-based polynomials

24 Pages·2017·0.39 MB·English

Note di Matematica ISSN 1123-2536, e-ISSN 1590-0932
Note Mat. 37 (2017) no. 2, 21–43. doi:10.1285/i15900932v37n2p21
Certain generating funtion of generalized
Apostol type Legendre-based polynomials
N.U. Khan
Department of Applied Mathematics, Faculty of Engineering and Technology,
Aligarh Muslim University, Aligarh-202002, India.
[email protected]
T. Usman∗
Department of Applied Mathematics, Faculty of Engineering and Technology,
Aligarh Muslim University, Aligarh-202002, India.
[email protected]
M. Aman
Department of Applied Mathematics, Faculty of Engineering and Technology,
Aligarh Muslim University, Aligarh-202002, India.
[email protected]
Received: 12.3.2017; accepted: 19.4.2017.
Abstract. Inthispaper,weaimtointroduceageneratingfunctionforgeneralizedApostol
type Legendre-Based polynomials which extends some known results. We also deduce some
properties of the generalized Apostol-Bernoulli polynomials, the generalized Apostol-Euler
polynomials and the generalized Apostol-Genocchi polynomials of higher order. By making
useofthegeneratingfunctionmethodandsomefunctionalequationsmentionedinthepaper,
we conduct a further investigation in order to obtain some implicit summation formulae and
general symmetry identities for the generalized Apostol type Legendre-Based polynomials.
Keywords: Hermite polynomials and their two variable extensions, generalized Apostol
Bernoulli numbers and polynomials, generalized Apostol Euler numbers and polynomials,
generalized Apostol Genocchi numbers and polynomials, Legendre polynomials and their two
variableextensions,0thorderTricomifunction,generalizedApostoltypeLegendre-Basedpoly-
nomials, summation formulae, symmetric identities.
MSC 2000 classification: 5A10, 11B68, 33C45, 33C99.
1 Introduction
Generalized and multivariable forms of the special functions of mathematical
physics have witnessed a significant evolution during the recent years. In par-
ticular, the special polynomials of more than one variable provided new means
of analysis for the solution of large classes of partial differential equations often
encountered in physical problems. Most of the special function of mathematical
http://siba-ese.unisalento.it/ (cid:13)c 2017 Universit`a del Salento
22 N.U. Khan, T. Usman and M. Aman
physics and their generalization have been suggested by physical problems.
To give an example, we recall that the 2-variable Hermite Kampe(cid:48) de Fe(cid:48)riet
polynomials H (x,y) [2] defined by the generating function
n
(cid:88)∞ tn
exp(xt+yt2) = H (x,y) (1.1)
n
n!
n=0
are the solution of heat equation
∂ ∂2
H (x,y) = H (x,y), H (x,0) = xn (1.2)
∂y n ∂x2 n n
The higher order Hermite polynomials, sometimes called the Kampe(cid:48) de Fe(cid:48)riet
(m)
polynomials of order m or the Gould-Hopper polynomials H (x,y) defined by
n
the generating function ([11], p.58 (6.3))
(cid:88)∞ tn
exp(xt+ytm) = H(m)(x,y) (1.3)
n
n!
n=0
are the solution of the generalized heat equation [7]
∂ ∂m
f(x,y) = f(x,y), f(x,0) = xn (1.4)
∂y ∂xm
Also, we note that
H(2)(x,y) = H (x,y), (1.5)
n n
H (2x,−1) = H (x), (1.6)
n n
where H (x) are the classical Hermite polynomials [1].
n
Next, we recall that the 2-variable Legendre polynomials S (x,y) and R (x,y)
n n
are given by Dattoli et al. [8]
[n]
(cid:88)2 xkyn−2k
S (x,y) = n! (1.7)
n [(n−2k)!(k!)2]
k=0
and
(cid:88)∞ (−1)n−kxn−kyk
R (x,y) = (n!)2 (1.8)
n [(n−2k)!]2(k!)2
k=0
respectively, and are related with the ordinary Legendre polynomials P (x)
n
[27] as
Certain generating function of generalized Apostol type Legendre-based polynomials23
(cid:18) 1−x2 (cid:19) (cid:18)1−x 1+x(cid:19)
P (x) = S − ,x = R , .
n n n
4 2 2
From equation (7) and (8), we have
S (x,0) = n! x[n2] , S (0,y) = yn, (1.9)
n [(n)!]2 n
2
R (x,0) = (−x)n, R (0,y) = yn. (1.10)
n n
The generating functions for two variable Legendre polynomials S (x,y) and
n
R (x,y) are given by [8]
n
(cid:88)∞ tn
eytC (−xt2) = S (x,y) (1.11)
0 n
n!
n=0
(cid:88)∞ tn
C (xt)C (−yt) = R (x,y) (1.12)
0 0 n (n!)2
n=0
where C (x) is the 0-th order Tricomi function [27]
0
(cid:88)∞ (−1)rxr
C (x) = (1.13)
0 (r!)2
r=0
(α)
The generalized Bernoulli polynomials B (x) of order α ∈ C, the gen-
n
(α)
eralized Euler polynomials E (x) of order α ∈ C and generalized Genocchi
n
(α)
polynomials G (x) of order α ∈ C, are defined respectively by the following
n
generating functions (see [10], vol. 3, p. 253 et seq.), ([16], section 2.8) and [18]):
(cid:18) t (cid:19)α (cid:88)∞ tn
ext = B(α)(x) , (|t| < 2π;1α := 1) (1.14)
et−1 n n!
n=0
(cid:18) 2 (cid:19)α (cid:88)∞ tn
ext = E(α)(x) , (|t| < π;1α := 1) (1.15)
et+1 n n!
n=0
(cid:18) 2t (cid:19)α (cid:88)∞ tn
ext = G(α)(x) , (|t| < π;1α := 1) (1.16)
et+1 n n!
n=0
The literature contains a large number of interesting properties and relation-
shipinvolvingthesepolynomials(see[3],[4],[5],[10],[12],[24]).LuoandSrivastava
24 N.U. Khan, T. Usman and M. Aman
(α)
([20],[22]) introduced the generalized Apostol Bernoulli polynomials B (x;λ)
n
(α)
of order α, Luo [17] investigated Apostol Euler polynomials E (x;λ) of order
n
(α)
α and the generalized Apostol Genocchi polynomials G (x;λ) of order α (see
n
also [18],[19],[21]).
(α)
The generalized Apostol Bernoulli polynomials B (x;λ) of order α ∈ C,
n
(α)
the generalized Apostol Euler polynomials E (x;λ) of order α ∈ C and gen-
n
(α)
eralized Apostol Genocchi polynomials G (x;λ) of order α ∈ C, are defined
n
respectively by the following generating functions:
(cid:18) t (cid:19)α (cid:88)∞ tn
ext = B(α)(x;λ) , (|t+ln λ| < 2π;1α := 1) (1.17)
λet−1 n n!
n=0
(cid:18) 2 (cid:19)α (cid:88)∞ tn
ext = E(α)(x;λ) , |t+ln λ| < π;1α := 1) (1.18)
λet+1 n n!
n=0
(cid:18) 2t (cid:19)α (cid:88)∞ tn
ext = G(α)(x;λ) , (|t+ln λ| < π;1α := 1) (1.19)
λet+1 n n!
n=0
where, if we take x = 0 in the above, we have
B(α)(0;λ) := B(α)(λ), E(α)(0;λ) := E(α)(λ), G(α)(0;λ) = G(α)(λ) (1.20)
n n n n n n
calling Apostol-Bernoulli number of order α, Apostol-Euler number of order α
and Apostol-Genocchi number of order α, respectivily. Also
B(α)(x) := B(α)(x;1), E(α)(x) := E(α)(x;1), G(α)(x) = G(α)(x;1). (1.21)
n n n n n n
Srivastava et al. [29],[30] have investigated the new class of generalized Apostol-
(α)
Bernoulli polynomials B (x;λ;a,b,e) of order α, Apostol-Euler polynomials
n
(α) (α)
E (x;λ;a,b,e) of order α and Apostol-Genocchi polynomials G (x;λ;a,b,e)
n n
of order α, are defined respectively by the following generating functions:
(cid:18) t (cid:19)α (cid:88)∞ tn (cid:16)a(cid:17)
ext = B(α)(x;λ;a,b,e) , (|t ln +ln λ| < 2π;1α := 1)
λbt−at n n! b
n=0
(1.22)
Certain generating function of generalized Apostol type Legendre-based polynomials25
(cid:18) 2 (cid:19)α (cid:88)∞ tn (cid:16)a(cid:17)
ext = E(α)(x;λ;a,b,e) , (|t ln +ln λ| < π;1α := 1)
λbt+at n n! b
n=0
(1.23)
(cid:18) 2t (cid:19)α (cid:88)∞ tn (cid:16)a(cid:17)
ext = G(α)(x;λ;a,b,e) , (|t ln +ln λ| < π;1α := 1)
λbt+at n n! b
n=0
(1.24)
If we take a = 1, b = e in (22), (23) and (24) respectively, we have (17), (18)
and (19). Obviously when we set λ = 1, α = 1, b = e in (22), (23) and (24), we
have classical Bernoulli polynomials B (x), classical Euler polynomials E (x)
n n
and classical Genocchi polynomials G (x).
n
Recently, Luo et al. [23] introduced a generalized Apostol type polynomials
(α)
F (x;λ;µ,ν) (α ∈ N ,µ,ν ∈ C) of order α, are defined by means of the
n 0
following generating function:
(cid:18) 2µtν (cid:19)α (cid:88)∞ tn
ext = F(α)(x;λ;µ,ν) , |t| < |log(−λ)) (1.25)
λet+1 n n!
n=0
where
F(α)(λ;µ,ν) = F(α)(0;λ;µ,ν) (1.26)
n n
denotes the so called Apostol type number of order α.
So that by comparing equation (17), (18) and (19), we have
B(α)(x;λ) = (−1)αF(α)(x;−λ;0,1) (1.27)
n n
E(α)(x;λ) = F(α)(x;λ;1,0) (1.28)
n n
G(α)(x;λ) = F(α)(x;λ;1,1) (1.29)
n n
Thespecialpolynomialsofmorethanonevariableprovidenewmeansofanalysis
for the solutions of a wide class of partial differential equations often encoun-
tered in physical problems. It happens very often that the solution of a given
problem in physics or applied mathematics requires the evaluation of infinite
sum, involving special functions. Problem of this type arise, for example, in the
computation of the higher-order moments of a distribution or in evaluation of
26 N.U. Khan, T. Usman and M. Aman
transition matrix elements in quantum mechanics. In [6], Dattoli showed that
the summation formulae of special functions, often encountered in applications
ranging from electromagnetic process to combinatorics, can be written in terms
of Hermite polynomials of more than one variable.
In this paper, we first give definition of the generalized Apostol type Legendre-
(α)
Based polynomials F (x,y,z;λ;µ,ν) which generalizes the concept stated
S n
above and then find their basic properties and relationships with Apostol type
(α)
Hermite-Based polynomials F (x,y;λ;µ,ν) of Lu et al. [23]. Some implicit
H n
summationformulaeandgeneralsymmetryidentitiesarederivedbyusingdiffer-
ent analytical means and applying generating functions. These result extends
some known summation and identities of generalized Apostol type Hermite-
Bernoulli, Euler and Genocchi polynomials studied by Dattoli et al. [9], Yang
[31], Khan et al. [13]-[15], Pathan [25], Pathan and Khan [26], Yang et al. [32]
and Zhang et al. [33].
2 Definition and Properties of the Generalized Apos-
tol type Legendre-Based polynomials
In this section, we present further definition and properties for the generalized
(α)
Apostol type Legendre-Based polynomials F (x,y,z;λ;µ,ν).
S n
Definition 2.1. The generalized Apostol type Legendre-Based polynomials
(α)
F (x,y,z;λ;µ,ν) (α ∈ N ,µ,ν ∈ C) for nonnegative integer n, are defined
S n 0
by
(cid:88)∞ F(α)(x,y,z;λ;µ,ν)tn = (cid:18) 2µtν (cid:19)αeyt+zt2C (−xt2), (|t| < |log(−λ))
S n n! λet+1 0
n=0
(2.1)
so that
F(α)(x,y,z;λ;µ,ν) = (cid:88)n (cid:88)[m2] Fn(α−)m(λ;µ,ν) Sm−2k(x,y)zkn! (2.2)
S n (m−2k)!k!(n−m)!
m=0k=0
For α = 1, in (30) we obtain the following generating function
(cid:88)∞ F (x,y,z;λ;µ,ν)tn = (cid:18) 2µtν (cid:19)eyt+zt2C (−xt2), (|t| < |log(−λ))
S n n! λet+1 0
n=0
(2.3)
Certain generating function of generalized Apostol type Legendre-based polynomials27
For x = 0 in (30), the result reduces to Hermite-Based generalized Apostol
type polynomials of Lu et al. [23] is defined as
(cid:88)∞ F(α)(y,z;λ;µ,ν)tn = (cid:18) 2µtν (cid:19)αeyt+zt2, (|t| < |log(−λ)) (2.4)
H n n! λet+1
n=0
As in the case y = z = 0 in (30), it leads to an extension of the generalized
(α)
Apostol type polynomials denoted by F (x;λ;µ,ν) for a nonnegative integer
n
n defined earlier by (25).
(α)
ThegeneralizedApostoltypeLegendre-Basedpolynomials F (x,y,z;λ;µ,ν,e)
S n
definedby(30)havethefollowingpropertieswhicharestatedastheorembelow.
Theorem 2.1. For any integral n ≥ 1, x,y,z ∈ R, λ ∈ C and α ∈ N. The
following relation for the generalized Apostol type Legendre-Based polynomials
(α)
F (x,y,z;λ;µ,ν) holds true:
S n
F(α)(x,y,z;λ;µ,ν,e) = F(α)(x,y,z;λ;µ,ν),
S n S n
(−1)α F(α)(x,y,z;−λ;0,1) = B(α)(x,y,z;λ)
S n S n
(2.5)
F(α)(x,y,z;λ;1,0) = E(α)(x,y,z;λ),
S n S n
F(α)(x,y,z;λ;1,1) = G(α)(x,y,z;λ)
S n S n
F(α+β)(x,y+z,v+u;λ;µ,ν) =
S n
n (cid:18) (cid:19)
(cid:88) n (α) (β)
F (x,z,v;λ;µ,ν) F (y,u;λ;µ,ν) (2.6)
k S n−k H k
k=0
F(α+β)(x,y+v,z;λ;µ,ν) =
S n
n (cid:18) (cid:19)
(cid:88) n (α) (β)
F (x,y,z;λ;µ,ν)F (v;λ;µ,ν) (2.7)
k S n−k k
k=0
Proof. The proof of (34) are obvious. Applying definition (30), we have
(cid:88)∞ tn
F(α+β)(x,y+z,v+u;λ;µ,ν)
S n
n!
n=0
= (cid:32)(cid:88)∞ F(α)(x,z,v;λ;µ,ν)tn(cid:33)(cid:32)(cid:88)∞ F(β)(y,u;λ;µ,ν)tk(cid:33)
S n n! H k k!
n=0 k=0
28 N.U. Khan, T. Usman and M. Aman
(cid:88)∞ (cid:32)(cid:88)n (cid:18) n (cid:19) (α) (β) (cid:33) tn
= F (x,z,v;λ;µ,ν) F (x,y,u;λ;µ,ν)
k S n−k H k n!
n=0 k=0
Now equating the coefficient of tn in the above equation, we get the result (35).
n!
Again by definition (30) of Apostol type Legendre-Based polynomials, we have
(cid:88)∞ F(α+β)(x,y+v,z;λ;µ,ν)tn = (cid:18) 2µtν (cid:19)α+βe(y+v)t+zt2C (−xt2)
S n n! λet+1 0
n=0
(cid:32) (cid:33)
(cid:18)(cid:18) 2µtν (cid:19)α (cid:19) (cid:18) 2µtν (cid:19)β
= eyt+zt2C (−xt2) evt
λet+1 0 λet+1
which can be written as
= (cid:88)∞ F(α)(x,y,z;λ;µ,ν)tn (cid:88)∞ F(β)(v;λ;µ,ν)tk
S n n! k k!
n=0 k=0
(cid:88)∞ (cid:32)(cid:88)n (cid:18) n (cid:19) (α) (β) (cid:33) tn
= F (x,y,z;λ;µ,ν)F (v;λ;µ,ν)
k S n−k k n!
n=0 k=0
Now equating the coefficient of the like power of tn in the above equation, we
n!
get the result (36).
3 Implicit Summation Formulae Involving Apostol
type Legendre-Based Polynomials
ForthederivationofimplicitformulaeinvolvinggeneralizedApostoltypeLegendre-
(α)
Basedpolynomials F (x,y,z;λ;µ,ν)thesameconsiderationasdevelopedfor
S n
the ordinary Hermite and related polynomials in Khan et al. [14] and Hermite-
Bernoulli polynomials in Pathan [25], Pathan and Khan [26] and Khan et al.
[13]-[15] holds as well. First we prove the following results involving generalized
(α)
Apostol type Legendre-Based polynomials F (x,y,z;λ;µ,ν).
S n
Theorem 3.1. For any integral n ≥ 1, x,y,z ∈ R, λ ∈ C and α ∈ N.
The following implicit summation formulae for the generalized Apostol type
(α)
Legendre-Based polynomials F (x,y,z;λ;µ,ν) holds true:
S n
(α)
F (x,v,z;λ;µ,ν) =
S m+n
m,n (cid:18) (cid:19)(cid:18) (cid:19)
(cid:88) m n (v−y)s+k F(α) (x,v,z;λ;µ,ν) (3.1)
s k S m+n−s−k
s,k=0
Certain generating function of generalized Apostol type Legendre-based polynomials29
Proof. We replace t by t+u and rewrite the generating function (30) as
(cid:18)2µ(t+u)ν(cid:19)α
ez(t+u)2C (−x(t+u)2)
λet+u+1 0
= e−y(t+u) (cid:88)∞ F(α) (x,y,z;λ;µ,ν) tn um (3.2)
S m+n n! m!
m,n=0
Replacing y by v in the above equation and equating the resulting equation to
the above equation, we get
e(v−y)(t+u) (cid:88)∞ F(α) (x,y,z;λ;µ,ν)tnum
S m+n n! m!
m,n=0
(cid:88)∞ (α) tnum
= F (x,v,z;λ;µ,ν) (3.3)
S m+n n! m!
m,n=0
On expanding exponential function (39) gives
(cid:88)∞ [(v−y)(t+u)]N (cid:88)∞ (α) tnum
F (x,y,z;λ;µ,ν)
N! S m+n n! m!
N=0 m,n=0
(cid:88)∞ (α) tnum
= F (x,v,z;λ;µ,ν) (3.4)
S m+n n! m!
m,n=0
which on using the following formula ([28], p. 52(2))
(cid:88)∞ (x+y)N (cid:88)∞ xnym
f(N) = f(m+n) (3.5)
N! n! m!
N=0 n,m=0
in the left hand side becomes
(cid:88)∞ (v−y)k+s tkus (cid:88)∞ (α) tnum
F (x,y,z;λ;µ,ν)
k!s! S m+n n! m!
k,s=0 m,n=0
(cid:88)∞ (α) tnum
= F (x,v,z;λ;µ,ν) (3.6)
S m+n n! m!
m,n=0
Now replacing n by n−k, s by n−s and using the lemma ([28], p. 100(1))
in the left hand side of (42), we get
(cid:88)∞ (cid:88)m,n (v−y)k+s (α) tn um
F (x,y,z;λ;µ,ν)
k!s! S m+n−k−s (n−k)!(m−s)!
m,n=0k,s=0
30 N.U. Khan, T. Usman and M. Aman
(cid:88)∞ (α) tnum
= F (x,v,z;λ;µ,ν) (3.7)
S m+n n! m!
m,n=0
Finally, on equating the coefficient of the like powers of tn and um in the
above equation, we get the required result.
Remark 3.1.1. Replacing λ = −λ, µ = 0 and ν = 1 in Theorem (3.1) and
then multiplying (−1)α on both side of the result, we immediately deduce the
following corollary.
Corollary 3.1.1.Thefollowingimplicitsummationformulaforthegeneralized
(α)
Apostol type Legendre-Bernoulli polynomials B (x,y,z;λ) holds true:
S n
m,n (cid:18) (cid:19)(cid:18) (cid:19)
B(α) (x,v,z;λ) = (cid:88) m n (v−y)s+k B(α) (x,v,z;λ) (3.8)
S m+n s k S m+n−s−k
s,k=0
Remark 3.1.2. By taking µ = 1 and ν = 0 in Theorem (3.1), we immediately
deduce the following corollary.
Corollary 3.1.2.Thefollowingimplicitsummationformulaforthegeneralized
(α)
Apostol type Legendre-Euler polynomials E (x,y,z;λ) holds true:
S n
m,n (cid:18) (cid:19)(cid:18) (cid:19)
E(α) (x,v,z;λ) = (cid:88) m n (v−y)s+k E(α) (x,v,z;λ) (3.9)
S m+n s k S m+n−s−k
s,k=0
Remark 3.1.3. By taking µ = 1 and ν = 1 in Theorem (3.1), we immediately
deduce the following corollary.
Corollary 3.1.3.Thefollowingimplicitsummationformulaforthegeneralized
(α)
Apostol type Legendre-Genocchi polynomials G (x,y,z;λ) holds true:
S n
(α)
G (x,v,z;λ) =
S m+n
m,n (cid:18) (cid:19)(cid:18) (cid:19)
(cid:88) m n (v−y)s+k G(α) (x,v,z;λ) (3.10)
s k S m+n−s−k
s,k=0
Theorem 3.2. For any integral n ≥ 1, x,y,z ∈ R, λ ∈ C and α ∈ N. The
followingimplicitsummationformulaforthegeneralizedApostoltypeLegendre-
(α)
Based polynomials F (x,y,z;λ;µ,ν) holds true:
S n
n (cid:18) (cid:19)
F(α)(x,y+u,z;λ;µ,ν) = (cid:88) n uj F(α)(x,y,z;λ;µ,ν) (3.11)
S n j S n−j
j=0