Generalized Incomplete Pochhammer Symbols and Their Applications to Hypergeometric Functions

Vivek Sahai; Ashish Verma

doi:10.5666/KMJ.2018.58.1.67

Article

Kyungpook Mathematical Journal 2018; 58(1): 67-79

Published online March 23, 2018

Generalized Incomplete Pochhammer Symbols and Their Applications to Hypergeometric Functions

Vivek Sahai, and Ashish Verma^*

Department of Mathematics and Astronomy, Lucknow University, Lucknow 226007, India, e-mail : sahai_vivek@hotmail.com and vashish.lu@gmail.com

Received: February 10, 2017; Accepted: February 13, 2018

Abstract

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Abstract
1. Introduction
2. Generalized Incomplete Pochhammer Symbols
3. Generating Functions and Reduction Formulas
4. Conclusion
References

In this paper, we present new generalized incomplete Pochhammer symbols and using this we introduce the extended generalized incomplete hypergeometric functions. We derive certain properties, generating functions and reduction formulas of these extended generalized incomplete hypergeometric functions. Special cases of this extended generalized incomplete hypergeometric functions are also discussed.

Keywords: incomplete gamma functions, incomplete Pochhammer symbols, generalized incomplete hypergeometric functions, Generating functions

1. Introduction

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Abstract
1. Introduction
2. Generalized Incomplete Pochhammer Symbols
3. Generating Functions and Reduction Formulas
4. Conclusion
References

In a recent paper, Srivastava et al. [10] have studied incomplete Pochhammer symbols and generalized incomplete hypergeometric functions associated with them. It was shown that these generalized incomplete hypergeometric functions have applications in areas such as communication theory, probability theory and groundwater pumping modeling. In the present paper, we introduce a new pair of generalized incomplete Pochhammer symbols and study the resulting extended generalized incomplete hypergeometric functions from the generating functions and reduction formulas point of view. We start with recalling the necessary definitions and results.

The incomplete gamma functions γ(s, x) and Γ(s, x) are defined by [11]

(1.1)γ(s,x)=∫0xts-1e-tdt, ℜ(s)≥0;x≥0

and

(1.2)Γ(s,x)=∫x∞ts-1e-tdt, ℜ(s)≥0;x≥0

respectively and satisfy the following decomposition formula:

(1.3)γ(s,x)+Γ(s,x)=Γ(s), ℜ(s)>0.

Srivastava et al. [10] introduced the generalized incomplete hypergeometric functions by

(1.4)γrs [(a1,x),a2,…,ar;b1,…,bs;z]=∑n=0∞(a1;x)n(a2)n⋯(ar)n(b1)n⋯(bs)nznn!

and

(1.5)Γrs [(a1,x),a2,…,ar;b1,…,bs;z]=∑n=0∞[a1;x]n(a2)n⋯(ar)n(b1)n⋯(bs)nznn!,

where (a1;x)n=γ(a1+n,x)Γ(a) and [a1;x]n=Γ(a1+n,x)Γ(a) are incomplete Pochhammer symbols. Note that _rγ_s and _rΓ_s satisfy the decomposition formula:

(1.6)γrs [(a1,x),a2,…,ar;b1,…,bs;z]+Γrs [(a1,x),a2,…,ar;b1,…,bs;z]=Frs [a1,…,ar;b1,…,bs;z]

where _rF_s is the generalized hypergeometric function.

The generalized incomplete gamma functions γ(λ, x; α) and Γ(λ, x; α) are defined as [2]:

(1.7)γ(λ,x;α)=∫0xtλ-1 exp(-t-αt)dt, for α=0, ℜ(λ)>0,

and

(1.8)Γ(λ,x;α)=∫x∞tλ-1 exp(-t-αt) dt

respectively. These generalized incomplete gamma functions satisfy the following decomposition relation:

(1.9)γ(λ,x;α)+Γ(λ,x;α)=2αλ2Kλ(2α),

where K_λ(z) is a modified Bessel function of third kind, also known as Macdonald function, [15].

R. Srivastava [12, 13] and Srivastava and Cho [14] investigated several properties of the incomplete hypereometric functions and some general classes of the incomplete hypergeometric polynomials associated with them.

In [7], the authors have studied the following generalization of the Pochhammer symbol given by

(1.10)(λ;p,α,β)ν=Γp(α,β)(λ+ν)Γ(λ), ℜ(α)>0, ℜ(β)>0, ℜ(p)>0, λ,ν∈ℂ,

where Γp(α,β)(x) is a generalization of gamma function given by [5]

(1.11)Γp(α,β)(x)=∫0∞tx-1 1F1 (α;β;-t-pt) dt.

Note that for α = β, Γp(α,α)(x)=Γp(x), [1, 2], and (λ;p,α,α)ν=(λ;p)ν=Γp(λ+ν)Γp(ν), [9]. Clearly for p = 0, Γ_p(x) = Γ(x), the gamma function.

Section-wise treatment is as follows. In Section 2, we define a new pair of generalized incomplete Pochhammer symbols and study the resulting extended generalized incomplete hypergeometric functions. In Section 3, we obtain generating functions for the families of extended generalized incomplete hypergeometric functions. Also, we present a theorem that gives reduction formulas for extended generalized incomplete hypergeometric functions.

2. Generalized Incomplete Pochhammer Symbols

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Abstract
1. Introduction
2. Generalized Incomplete Pochhammer Symbols
3. Generating Functions and Reduction Formulas
4. Conclusion
References

We now introduce a pair of new extended incomplete gamma functions denoted by γ(λ, x; p, α, β) and Γ(λ, x; p, α, β) and defined as follows:

(2.1)γ(λ,x;p,α,β)=∫0xtλ-1 1F1 (α;β;-t-pt) dt

and

(2.2)Γ(λ,x;p,α,β)=∫x∞tλ-1 1F1 (α;β;-t-pt) dt,

where ℜ(α) > 0, ℜ(β) > 0, ℜ(p) > 0, ℜ(λ) > 0. These extended incomplete gamma functions satisfy the decomposition relation:

(2.3)γ(λ,x;p,α,β)+Γ(λ,x;p,α,β)=Γp(α,β)(λ).

Using these extended incomplete gamma functions γ(λ, x; p, α, β) and Γ(λ, x; p, α, β), we introduce a pair of new generalized incomplete Pochhammer symbols (λ, x; p, α, β)_ν and [λ, x; p, α, β]_ν, λ, ν ∈ ℂ, x ≥ 0, defined by:

(2.4)(λ,x;p,α,β)ν=γ(λ+ν,x;p,α,β)Γ(λ), λ,ν∈ℂ, x≥0,

and

(2.5)[λ,x;p,α,β]ν=Γ(λ+ν,x;p,α,β)Γ(λ), λ,ν∈ℂ, x≥0.

It is evident that the generalized incomplete Pochhammer symbols (2.4) and (2.5) satisfy the following decomposition formula

(2.6)(λ,x;p,α,β)ν+[λ,x;p,α,β]ν=(λ;p,α,β)ν, λ,ν∈ℂ, x≥0,

In view of (2.6), it is sufficient to study the properties of the [λ, x; p, α, β]_ν.

Motivated by the generalization of the incomplete Pochhammer symbols (2.4) and (2.5), we introduce the following extended generalized incomplete hypegeometric functions

(2.7)γrs(p,α,β) [(a1,x,p,α,β),a2,…,ar;b1,…,bs;z]=∑n=0∞(a1,x;p,α,β)n(a2)n…(ar)n(b1)n…(bs)nznn!

and

(2.8)Γrs(p,α,β) [(a1,x,p,α,β),a2,…,ar;b1,…,bs;z]=∑n=0∞[a1,x;p,α,β]n(a2)n…(ar)n(b1)n…(bs)nznn!.

From (2.7) and (2.8), we get the following decomposition formula

(2.9)γrs(p,α,β) [(a1,x,p,α,β),a2,…,ar;b1,…,bs;z]+Γrs(p,α,β) [(a1,x,p,α,β),a2,…,ar;b1,…,bs;z]=Frs [(a1,p,α,β),a2,…,ar;b1,…,bs;z]

where the right hand side is defined in [7]. Decomposition formula (2.9) is based on the decomposition formula (2.6), and generalizes (1.3).

We now discuss the case α = β of (2.4) and (2.5). This leads to the pair of generalized incomplete Pochhammer symbols (λ, x; p)_ν and [λ, x; p]_ν defined as follows:

(2.10)(λ,x;p)ν=γ(λ+ν,x;p)Γ(λ), λ,ν∈ℂ;x≥0,

and

(2.11)[λ,x;p]ν=Γ(λ+ν,x;p)Γ(λ), λ,ν∈ℂ;x≥0.

Clearly (λ, x; p)_ν + [λ, x; p]_ν = (λ; p)_ν. These incomplete Pochhammer symbols (2.10) and (2.11) lead to the following extended generalized incomplete hypegeometric functions

(2.12)γrs(p) [(a1,x,p),a2,…,ar;b1,…,bs;z]=∑n=0∞(a1,x;p)n(a2)n…(ar)n(b1)n…(bs)nznn!

and

(2.13)Γrs(p) [(a1,x,p),a2,…,ar;b1,…,bs;z]=∑n=0∞[a1,x;p]n(a2)n…(ar)n(b1)n…(bs)nznn!,

respectively. From (2.12) and (2.13), we get the decomposition formula

(2.14)γrs(p) [(a1,x,p),a2,…,ar;b1,…,bs;z]+Γrs(p) [(a1,x,p),a2,…,ar;b1,…,bs;z]=Frs [(a1,p),a2,…,ar;b1,…,bs;z]

where the right hand side is defined in [9]. Note that the decomposition formula (2.14) generalizes (1.6). Since |(λ; x, p)_n| ≤ |(λ)_n| and |[λ; x, p]_n| ≤ |(λ)_n|, n ∈ ℕ₀; λ ∈ ℂ; x ≥ 0 the infinite series in (2.12) and (2.13) converges absolutely. This can be verified from comparing these series with the case of the generalized hypergeometric function _rF_s, [6].

We now give the following results for the extended generalized incomplete hypergeometric function _rΓ_s(p, α, β), r,s ∈ ℕ₀. Analogous properties for the generalized incomplete hypergeometric function _rγ_s(p, α, β), r, s ∈ ℕ₀ can be derived using (2.9).

Theorem 2.1

The following integral representation holds true:

(2.15)Γrs(p,α,β) [(a1,x,p,α,β),a2,…,ar;b1,…,bs;z]=1Γ(a1)∫x∞ta1-1 1F1 (α;β;-t-pt) r-1Fs [a2,…,ar;b1,…,bs;zt] dt, ℜ(α)>0, ℜ(β)>0, ℜ(p)>0; ℜ(a1)>0, α=β when p=0.

Proof

Putting the generalized incomplete Pochhammer symbol [a₁, x; p, α, β]_n in (2.8) by its integral representation (2.2), we get the desired result.

Corollary 2.2

The following integral representation holds true:

(2.16)Γ21(p,α,β) [(a,x,p,α,β),b;c;z]=1Γ(a)∫x∞ta-1 1F1 (α;β;-t-pt) 1F1 (b;c;zt) dt, ℜ(α)>0, ℜ(β)>0, ℜ(p)>0,x≥0; ℜ(a)>0, α=β when p=0.

Theorem 2.3

The following derivative formula holds true:

(2.17)dndzn {Γrs(p,α,β) [(a1,x,p,α,β),a2,…,ar;b1,…,bs;z]}=∏i=1r(ai)n∏j=1s(bj)n rΓs(p,α,β) [(a1+n,x,p,α,β),a2+n,…ar+n;b1+n,…,bs+n;z], n∈ℕ0.

Proof

Obviously (2.17) is valid for n = 0.

We first prove (2.17) for n = 1:

Differentiating both sides of (2.15) with respect to z and simplifying, we get

(2.18)ddz {Γrs(p,α,β) [(a1,x,p,α,β),a2,…,ar;b1,…,bs;z]}=∏i=1r(ai)1∏j=1s(bj)1 rΓs(p,α,β) [(a1+1,x,p,α,β),a2+1,…ar+1;b1+1,…,bs+1;z].

The general result is obtained by applying induction on n ∈ ℕ₀.

Theorem 2.4

The following integral representation holds true:

(2.19)Γrs(p,α,β) [(a1,x,p,α,β),a2,…,ar-1,b;b1,…,bs-1,c;z]=1B(b,c-b)∫01tb-1(1-t)(c-b)-1 r-1Γs-1(p,α,β) [(a1,x,p,α,β),a2,…,ar-1;b1,…,bs-1;zt] dtℜ(α)>0, ℜ(β)>0, ℜ(p)>0, ℜ(c)>ℜ(b)>0; x≥0.

Proof

The assertion of the theorem follows from:

(2.20)(b)n(c)n=B(b+n,c-b)B(b,c-b)=1B(b,c-b)∫01tb+n-1(1-t)c-b-1dt,ℜ(c)>ℜ(b)>0; n∈ℕ0.

We now present some theorems for extended incomplete Gauss hypergeometric function:

Theorem 2.5

The following recurrence relation holds true:

(2.21)[b-(c-1)]Γ21(p,α,β) [(a,x,p,α,β),b;c;z]=bΓ21(p,α,β) [(a,x,p,α,β),b+1;c;z]-(c-1)Γ21(p,α,β) [(a,x,p,α,β),b;c-1;z].

Proof

As the confluent hypergeometric function ₁F₁ satisfies the contiguous relation [6]:

(2.22)(c-b-1) F11 (b;c;z)=(c-1)F11 (b;c-1;z)-bF11 (b+1;c;z),

we have, using (2.16)

(2.23)(c-b-1)Γ21(p,α,β) [(a,x,p,α,β),b;c;z]=1Γ(a)∫x∞ta-1 F11 (α;β;-t-pt) [(c-1) F11 (b;c-1;zt)-b F11 (b+1;c;zt)] dt,

Now (2.23) can be written as

(2.24)[b-(c-1)]Γ21(p,α,β) [(a,x,p,α,β),b;c;z]=bΓ21(p,α,β) [(a,x,p,α,β),b+1;c;z]-(c-1)Γ21(p,α,β) [(a,x,p,α,β),b;c-1;z].

Theorem 2.6

The following recurrence relation holds true:

(2.25)azcΓ21(p,α,β) [(a+1,x,p,α,β),b+1;c+1;z]=Γ21(p,α,β) [(a,x,p,α,β),b+1;c;z]-Γ21(p,α,β) [(a,x,p,α,β),b;c;z].

Proof

Using (2.16), we can write the l.h.s. of (2.25)

(2.26)azΓ21(p,α,β) [(a+1,x,p,α,β),b+1;c+1;z]=aΓ(a+1)∫x∞ta-1 F11 (α;β;-t-pt) [zt F11 (p+1;c+1;zt)] dt.

Using the contiguous relation [6]:

(2.27)c F11 (b;c;z)-c F11 (b-1;c;z)=z F11 (b;c+1;z),

eq (2.26) can be written as

(2.28)az Γ21(p,α,β) [(a+1,x,p,α,β),b+1;c+1;z]=cΓ(a)∫x∞ta-1 F11 (α;β;-t-pt) [F11 (b+1;c;zt)-F11 (b;c;zt)] dt.

This gives

(2.29)azcΓ21(p,α,β) [(a+1,x,p,α,β),b+1;c+1;z]=Γ21(p,α,β) [(a,x,p,α,β),b+1;c;z]-Γ21(p,α,β) [(a,x,p,α,β),b;c;z].

Following theorem gives a relation between ₂Γ₁(p, α, β) and the incomplete gamma function γ(λ, x).

Theorem 2.7

The following integral representation holds true:

(2.30)Γ21(p,α,β) [(a,x,p,α,β),b;b+1;-z]=bz-aΓ(a)∫x∞t(a-b)-1 F11 (α;β;-t-pt) γ(b,zt) dt,ℜ(α)>0, ℜ(β)>0, ℜ(p)>0; ℜ(a)>0 when α=β, p=0.

Proof

Putting c = b + 1 and replacing z by −z in (2.16), we get

(2.31)Γ21(p,α,β) [(a,x,p,α,β),b;b+1;-z]=1Γ(a)∫x∞ta-1 F11 (α;β;-t-pt) F11 [b;b+1;-zt] dt.

Using ₁F₁[s, s + 1;−x] = sx⁻^sγ(s, x), we find that

(2.32)Γ21(p,α,β) [(a,x,p,α,β),b;b+1;-z]=bz-aΓ(a)∫x∞t(a-b)-1 F11 (α;β;-t-pt) γ(b,zt)dt.

This completes the proof.

3. Generating Functions and Reduction Formulas

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Abstract
1. Introduction
2. Generalized Incomplete Pochhammer Symbols
3. Generating Functions and Reduction Formulas
4. Conclusion
References

The main generating functions for the families of extended generalized incomplete hypergeometric functions are presented in the following theorem. For λ ∈ ℂ and N ∈ ℕ, let Δ(N, λ) denotes the following array of N parameters:

λN,λ+1N,…,λ+N-1N,

the array Δ(N, λ) being assumed to be empty when N = 0.

Theorem 3.1

The following generating function holds true for the families of extended generalized incomplete hypergeometric functions _rγ_s(p, α, β) and _rΓ_s(p, α, β):

(3.1)(1-t)-λγr+Ns(p,α,β) [Δ(N,λ),(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;z(1-t)N]=∑n=0∞(λ)nn!γr+Ns(p,α,β) [Δ(N,λ+n),(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;z] tn,∣t∣<1,λ∈ℂ, N∈ℕ,

and

(3.2)(1-t)-λΓr+Ns(p,α,β) [Δ(N,λ),(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;z(1-t)N]=∑n=0∞(λ)nn!Γr+Ns(p,α,β) [Δ(N,λ+n),(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;z] tn,∣t∣<,λ∈ℂ, N∈ℕ,

provided each member of (3.1) and (3.2) exists.

Proof

Using (2.8) and the identity

(3.3)∑n=0∞(λ)nn!zn=(1-z)-λ, ∣z∣<1;λ∈ℂ.

we get the generating function (3.1). Proof of (3.2) is similar.

Theorem 3.2

Each of the following generating function hold true for families of the extended generalized incomplete hypergeometric functions _rγ_s(p, α, β) and _rΓ_s(p, α, β) :

(3.4)(1-t)-λγr+Ns(p,α,β) [Δ(N,λ),(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;z (-t1-t)N]=∑n=0∞(λ)nn!γr+Ns(p,α,β) [Δ(N,-n),(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;z] tn,∣t∣<1,λ∈ℂ, N∈ℕ;

(3.5)(1-t)-λΓr+Ns(p,α,β) [Δ(N,λ),(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;z (-t1-t)N]=∑n=0∞(λ)nn!Γr+Ns(p,α,β) [Δ(N,-n),(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;z] tn,∣t∣<1,λ∈ℂ, N∈ℕ;

(3.6)(1-t)-λγr+2Ns(p,α,β) [Δ(2N,λ),(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;z (-4t(1-t)2)N]=∑n=0∞(λ)nn!γr+2Ns(p,α,β) [Δ(N,-n),Δ(N;λ+n),(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;z] tn,∣t∣<1,λ∈ℂ, N∈ℕ;

(3.7)(1-t)-λΓr+2Ns(p,α,β) [Δ(2N,λ),(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;z (-4t(1-t)2)N]=∑n=0∞(λ)nn!Γr+2Ns(p,α,β) [Δ(N,-n),Δ(N;λ+n),(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;z] tn,∣t∣<1,λ∈ℂ, N∈ℕ;

(3.8)(1-t)-λγrs(p,α,β) [(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;ztN]=∑n=0∞(λ)nn!γr+Ns+N(p,α,β) [Δ(N,-n),(a1,x,p,α,β),a2,⋯,ar;Δ(N,1-λ-n), b1,⋯,bs;z] tn,∣t∣<1,λ∈ℂ, N∈ℕ;

(3.9)(1-t)-λΓrs(p,α,β) [(a1,x,p,α,β),a2,⋯,ar;b1,⋯,bs;ztN]=∑n=0∞(λ)nn!Γr+Ns+N(p,α,β) [Δ(N,-n),(a1,x,p,α,β),a2,⋯,ar;Δ(N,1-λ-n), b1,⋯,bs;z] tn,∣t∣<1,λ∈ℂ, N∈ℕ;

provided that each member of equations (3.4) to (3.9) exists.

Proof

The proof of Theorem (3.2) is similar to the Theorem (3.1)

We remark that adding (3.1) and (3.2) and applying (2.9) leads to the result appearing in [7, Eq. 4.1], Similarly, adding (3.4) and (3.5) leads to [7, Eq. 4.3] and adding (3.6) and (3.7) leads to [7, Eq. 4.4] and adding (3.8) and (3.9) leads to [7, Eq. 4.5].

Theorem 3.3

The following reduction formulas hold true for extended generalized incomplete hypergeometric functions:

(3.10)γr+1s(p,α,β) [(a0,x,p,α,β),b1+m1,⋯,bt+mt,at+1,⋯,ar;b1,⋯,bt,bt+1,⋯,bs;z]=∑j1=0m1⋯∑jt=0mt∧(j1,⋯,jt)zJt×γr-t+1s-t(p,α,β) [(a0+Jt,x,p,α,β),at+1+Jt,⋯,ar+Jt;bt+1+Jt,⋯,bs+Jt;z],x≥0,t≤min(r,s),r,s∈ℕ0,r<s whenz∈ℂ,r=s when∣z∣<1,

and

(3.11)Γr+1s(p,α,β) [(a0,x,p,α,β),b1+m1,⋯,bt+mt,at+1,⋯,ar;b1,⋯,bt,bt+1,⋯,bs;z]=∑j1=0m1⋯∑jt=0mt∧(j1,⋯,jt) zJt×Γr-t+1s-t(p,α,β) [(a0+Jt,x,p,α,β),at+1+Jt,⋯,ar+Jt;bt+1+Jt,⋯,bs+Jt;z],x≥0,t≤min(r,s),r,s∈ℕ0,r<s whenz∈ℂ,r=s when∣z∣<1,

where, J_t = j₁ + ··· + j_t and

∧(j1,⋯.jt)=(m1j1)⋯(mtjt)(b2+m2)J1⋯(bt+mt)Jt-1(a0)Jt(at+1)Jt⋯(ar)Jt(b1)J1⋯(bt)Jt(bt+1)Jt⋯(bs)Jt.

Proof

The proof of (3.3) is based upon induction on t ∈ ℕ. We first prove (3.3) for t = 1, that is,

(3.12)γr+1s(p,α,β) [(a0,x,p,α,β),b1+m1,a2,⋯,ar;b1,⋯,bs;z]=∑j1=0m1(m1j1)(a0)j1(a2)j1⋯(ar)j1(b1)j1⋯(bs)j1zj1×γrs-1(p,α,β) [(a0+j1,x,p,α,β),a2+j1,⋯,ar+j1;b2+j1,⋯,bs+j1;z],x≥0, m1,r,s∈ℕ0, r<s when z∈ℂ, r=s when ∣z∣ <1.

Using the identity (m1j1)=(-1)j1(-m1)j1j1!, we write right hand side of (3.12) as

(3.13)∑j1=0m1(-1)j1(-m1)j1(a0)j1(a2)j1⋯(ar)j1j1!(b1)j1⋯(bs)j1zj1×γrs-1(α) [(a0+j1,x,p,α,β),a2+j1,⋯,ar+j1;b2+j1,⋯,bs+j1;z]=∑n=0∞∑j1=0m1(-1)j1(-m1)j1(a0)j1(a2)n+j1⋯(ar)n+j1(a0+j1,x;p,α,β)nj1!(b1)j1(b2)n+j1⋯(bs)n+j1zn+j1n!=∑n=0∞(a0,x;p,α,β)n(a2)n⋯(ar)n(b2)n⋯(bs)nznn!∑j1=0min{m1,n}(-m1)j1(-n)j1j1!(b1)j1=∑n=0∞(a0,x;p,α,β)n(b1+m1)n(a2)n⋯(ar)n(b1)n⋯(bs)nznn!,

using Chu-Vandermonde formula, [6]. This establishes (3.12). Inductively (3.10) is proved. The second reduction formula (3.11) can be proved in a similar manner.

4. Conclusion

Go to

Abstract
1. Introduction
2. Generalized Incomplete Pochhammer Symbols
3. Generating Functions and Reduction Formulas
4. Conclusion
References

We remark that our results on extended generalized incomplete hypergeometric functions generalize the corresponding results on generalized incomplete hypergeometric functions when α = β and p = 0, [10, 12, 13]. Further, for α = β, p = 0 and x = 0, the results of extended generalized incomplete hypergeometric functions _rΓ_s(p, α, β) reduce to the generalized hypergeometric functions _rF_s, [3, 4, 8].

References

Go to

Abstract
1. Introduction
2. Generalized Incomplete Pochhammer Symbols
3. Generating Functions and Reduction Formulas
4. Conclusion
References

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