A New Generalization of Pochhammer Symbol and Its Applications

Recep Şahin 1  and Oğuz Yağcı 1
  • 1 Department of Mathematics, Turkey
Recep Şahin
  • Department of Mathematics, Kırıkkale University, Faculty of Science Kırıkkale, Turkey
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and Oğuz Yağcı
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  • Department of Mathematics, Kırıkkale University, Faculty of Science Kırıkkale, Turkey
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Abstract

In this paper, we introduce a new generalization of the Pochhammer symbol by means of the generalization of extended gamma function . Using the generalization of Pochhammer symbol, we give a generalization of the extended hypergeo-metric functions one or several variables. Also, we obtain various integral representations, derivative formulas and certain properties of these functions.

1 Introduction

The classical Pochhammer symbol (λ)ν is given as follows: [1, 7, 9, 10, 18, 22, 23, 26, 29, 30, 33]

(λ)ν=Γ(λ+ν)Γ(λ)(λ, ν\0)={1λ(λ+1)(λ+n1)(ν=0)(ν=n).
and Γ(λ ) is the familiar Gamma function whose Euler’s integral is (see, e.g., [1, 7, 9, 10, 18, 22, 23, 30, 33])
Γ(z)=0 et tz1 dt((z)>0).
From (1) and (2), it is easy to see the following integral formula
(λ)ν=1Γ(λ) 0 et tλ+ν1 dt((λ+ν)>0).

Here and in the following, let ℂ, 0 , and ℕ be the sets of complex numbers, non-positive integers and positive integers, respectively and assume that min {ℜ(p), ℜ(q), ℜ(κ), ℜ(μ)} > 0. Recently, various generalization of beta functions have been introduced and investigated (see, e.g., [2, 3, 4, 5, 6, 9, 13, 14, 15, 16, 17, 19, 20, 21, 24, 25, 26, 28, 29] and the references cited therein). Very recently, Şahin et al. [31] introduced and studied following generalization of the extended gamma function as follows:

Γp,q(κ,μ)(z)= 0 tz1 exp(tκpqtμ)dt,((z)>0, (p)>0, (q)>0, (κ)>0, (μ)>0).
It is easily seen that the special cases of (4) returns to other forms of gamma functions. For example, Γ1,0(1,1)(x)= Γ(x) ,  Γ1,q(1,1)(x)= Γq(x) .

First, by selecting a known generalization of the gamma function (4), systematicaly, we goal to introduce new Pochhammer symbol by using the gamma function (4). Also, we give some properties for the extended Pochhammer symbol. Next, using this Pochhammer symbol, we define a new generalization of the extended hypergeometric functions one or several variables such as Gauss hypergeometric function, confluent hypergeometric function, Appell hypergeometric function, Humbert hypergeometric function. Finally, for this a new generalization of the extended hypergeometric functions, we give some properties such as integral representations, derivative formulas and recuurence relations.

2 A New Generalization of the Pochhammer Symbol

In this section, we denote a new generalization of Pochhammer symbol (5). Also, we give some useful properties.

Definition 1

Let λ, μ ∈ ℂ and ℜ(p) > 0, ℜ(q) > 0, ℜ(κ) > 0, ℜ(μ) > 0, the generalization of the extended Pochhammer symbol (λ; p,q;κ, μ)ν is given by

(λ;p,q;κ,μ)ν:={Γp,q(κ,μ)(λ+ν)Γ(λ), (p)>0, (q)>0, (κ)>0, (μ)>0,        (λ)ν, p=1, q=0, κ=1, μ=0,
where Γp,q(κ,μ) is the generalization of the extended gamma function (4) [31

Theorem 1

For the generalization of the Pochhammer symbol(5)following integral representation holds true:

(λ;p,q;κ,μ)ν:=1Γ(λ) 0 tλ+ν1 exp(tκpqtμ)dt((p)>0, (q)>0, (κ)>0, (μ)>0).

Proof

Using the equality (4) in the definition of the (5), we get the desired result (6).

Theorem 2

Let λ, m, n ∈ ℂ. Then,

(λ;p,q;κ,μ)n+m:=(λ)n(λ+n;p,q;κ,μ)m

Proof

From the equations (1) and (5), we obtain that

(λ;p,q;κ,μ)m+n:=Γp,q(κ,μ)(λ+m+n)Γ(λ)=Γ(λ+n)Γ(λ+n)Γp,q(κ,μ)(λ+m+n)Γ(λ)=(λ)n(λ+n;p,q;κ,μ)m

By appealing the well-known properties of the classical Pochhammer symbol in (7) following features of generalization of the Pochhammer symbol can be easily obtained.

Corollary 3

Let k,l,m,n ∈ ℕ0and N ℕ. Then,

(λ;p,q;κ,μ)m+n+l:=(λ)m(λ+m)n(λ+m+n;p,q;κ,μ)l(λ;p,q;κ,μ)mn+l:=(1)n(λ)m(1λm)n(λ+mn;p,q;κ,μ)l(λ;p,q;κ,μ)2m+l:=22m(λ2)m(λ+12)m(λ+2m;p,q;κ,μ)l(λ;p,q;κ,μ)Nm+l:=NNm(λN)m(λ+1N)m(λ+N1N)m(λ+Nm;p,q;κ,μ)l(λ+n;p,q;μ)n+1:=(λ+n)n(λ+2n;p,q;κ,μ)l=(λ)2n(λ)n(λ+2n;p,q,κ,μ)l(λ+m;p,q;κ,μ)n+l:=(λ)n(λ+n)m(λ)m(λ+m+n;p,q;κ,μ)l(λ+km;p,q;κ,μ)kn+l:=(λ)km+kn(λ)km(λ+km+kn;p,q;κ,μ)l(λn;p,q;κ,μ)n+1:=(1)n(1λ)n(λ;p,q;κ,μ)l(λm;p,q;κ,μ)n+1:=(1λ)m(λ)n(1λn)m(λ+n;p,q,κ,μ)l(λkm;p,q;κ,μ)kn+1:=(1)km(λ)knkm(1λ)km(λ+knkm;p,q,κ,μ)l(λ+m;p,q;κ,μ)nm+l:=(λ)n(λ)m(λ+n+;p,q;κ,μ)l(λm;p,q;κ,μ)nm+1:=(1)m(λ)n(1λ)m(1λn)2m(λ+n2m;p,q,κ,μ)l(λ;p,q;κ,μ)n+1:=(1)n(λn+1)(λ+n;p,q,κ,μ)l

Remark 1

Taking p = κ = μ = 1 in the Corollary 3, it is easily seen that the special case of extended Pochhammer symbol [29].

3 A New Generalization of the extended hypergeometric function

According to the generalization of the extended Pochhammer symbol (λ; p,q; κ, μ)n (n ∈ ℕ0), a generalization of the extended hypergeometric function rFs of r numerator parameters a1,⋯,ar and s denominator parameters b1,⋯,bs can be given as follows:

rFs[(a1;p,q;κ,μ),a2,  ,ar b1,b2,,bs ;z]:=n=0(a1;p,q;κ,μ)n(a2)n(ar)n(b1)n(bs)nznn!,
on condition that the series on the right-hand side converges, it making sense that aj ∈ ℂ ( j = 1,..., r) and bj\0(j=1,...,s; 0={0,1,2,...}) .

Particularly, the corresponding generalization of the extended confluent hypergeometric function Φp,qκ,μ and the Gauss hypergeometric function Fp,qκ,μ are given by

Φp,qκ,μ(a;b;z):= n=0 (a;p,q;κ,μ)n(b)nznn!
and
Fp,qκ,μ(a,b,c;z):=n=0 (a;p,q;κ,μ)n(b)n(c)nznn!,
respectively.

Theorem 4

The following integral representation holds true:

rFs[(a1;p,q;κ,μ),a2,  ,ar b1,b2,,bs ;z]:=1Γ(a1)0ta11exp(tκpqtμ)×r1Fs[a2,  ,ar b1,b2,,bs ;zt]dt,((p)>0, (q)>0, (κ)>0, (μ)>0; (bs)>(ar)>0,).

Proof

Using the integral representation given by (6) in the definition (9), we led to desired result (12).

Theorem 5

The following integral representation holds true:

rFs[(a1;p,q;κ,μ),a2,  ,ar b1,b2,,bs ;z]:=1B(ar,bsar)01 tar1(1t)bsar1× r1Fs1[(a1;p,q;κ,μ),a2,  ,ar1 b1,b2,,bs1 ;zt]dt,((p)>0, (q)>0, (κ)>0, (μ)>0; (bs)>(ar)>0).

Proof

The classical Beta function B(α,β ) defined by [1, 7, 9, 10, 18, 22, 23, 26, 29, 30, 33],

B(α,β)={01tα1(1t)β1dt(min{(α),(β)}>0)Γ(α)Γ(β)Γ(α+β)(α,β\0).                            
Also, we have the following equation
(ar)n(bs)n=1B(ar,bsar)01tar+n1(1t)bsar1dt,((bs)>(ar)>0;n0)

Using the equalities (14), (15) in the generization of the extended hypergeometric function (9), we get the desired result (13).

Corollary 6

Each of the integral representations hold true:

Φp,qκ,μ(a;b;z)=1Γ(a)0ta1exp(tκpqtμ) 0F1(;b;zt)dt,
Fp,qκ,μ(a,b,c;z)=1Γ(a)0ta1exp(tκpqtμ) 1F1(b;c;zt)dt,
and
Fp,qκ,μ(a,b,c;z)=1B(b,cb) 01tb1(1t)cb1 1F0((a;p,q;κ,μ);;zt)dt,
on condition that the integrals involved are convergent.

Theorem 7

The following derivative formula holds true:

dndzn{rFs[(a1;p,q;κ,μ),a2,  ,ar b1,b2,,bs ;z]}:=(a1)n(a2)n (ar)n(b1)n (bs)n×rFs[(a1+n;p,q;κ,μ),a2+n,  ,ar+n b1+n,b2+n,,bs+n ;z]((p)>0, (q)>0, (κ)>0, (μ)>0; n0).

Proof

Differentiating (9) with respect to z and then replacing nn + 1 in the right-hand side term, we obtain

ddz{rFs[(a1;p,q;κ,μ),a2,  ,ar b1,b2,,bs ;z]}:=n=0(a1;p,q;κ,μ)n+1(a2)n+1(ar)n+1(b1)n+1(b2)n+1(bs)n+1zn+1(n+1)=a1arb1bsrFs[(a1+1;p,q;κ,μ),a2+1,  ,ar+1 b1+1,b2+1,,bs+1 ;z],
repeating the same procedure n-times gives the formula (19).

Choosing r = s = 1 and r = 2, s = 1 in (19), we have the derivative formulas for the (10) and (11), respectively.

Corollary 8

The following derivative formulas hold true:

dndzn {Φp,qκ,μ(a;b;z)}=(a)n(b)n Φp,qκ,μ(a+n;b+n;z)
and
dndzn {Fp,qκ,μ(a,b,c;z)}=(a)n(b)n(c)n Fp,qκ,μ(a+n,b+n,c+n;z).

The Bessel function Jν(z) and the modified Bessel function Iν(z) are expressible as hypergeometric functions as follows [11, 12, 33]:

Jν(z)=(z2)νΓ(ν+1) 0F1(;ν+1;14z2)(ν \ (={1,2,3,...}))
and
Iν(z)=(z2)νΓ(ν+1) 0F1(;ν+1;14z2)(ν \ ).

Additionally, for the incomplete gamma function γ(s,x) defined by [10],

γ(s,x)=0xts1exp(t)dt((s)>0;x0).

Also, we know that [7, 9, 10]

1F1[s;s+1;x]=sxsγ(s,x).

So, we can deduce Corollary 9 and Corollary 10 by performing the relationships (23)(26) in the equations (16) and (17).

Corollary 9

Each of the following integral representations hold true:

Φp,qκ,μ(a;b+1;z)=Γ(b+1)Γ(a)zb20tab21exp(tκpqtμ)Jb(2zt)dt
and
Φp,qκ,μ(a;b+1;z)=Γ(b+1)Γ(a)zb20tab21exp(tκpqtμ)Ib(2zt)dt
on condition that the integrals involves are convergent.

Corollary 10

The following integral representation holds true:

Fp,qκ,μ(a,b,b+1;z)=bzbΓ(a)0tab1exp(tκpqtμ)γ(b,zt)dt,
on condition that the integrals involves are convergent.

4 A New Generalization of the extended Appell hypergeometric functions

In this section, we introduce extended Appell hypergeometric series and some extended multivariable hypergeometric functions.

Let us introduce the extensions of the Appell’s functions and extended Lauricella’s hypergeometric function and other functions defined by

p,qF1(κ,μ)[a,b,c;d;x,y]:= m,n=0(a;p,q;κ,μ)m+n (b)m (c)n(d)m+n xmm! ynn!max(|x|, |y|)< 1,
p,qF2(κ,μ)[a,b,c;d,e;x,y]:= m,n=0(a;p,q;κ,μ)m+n (b)m (c)n(d)m (e)n xmm! ynn!|x|+|y|< 1,
p,qF3(κ,μ)[a,b,c,d;e;x,y]:= m,n=0(a;p,q;κ,μ)m (b)n (c)m (d)n(e)m+n xmm! ynn!max(|x|, |y|)< 1,
p,qF4(κ,μ)[a,b;c,d;x,y]:= m,n=0(a;p,q;κ,μ)m+n (b)m+n(c)m (d)n xmm! ynn!|x|+|y|< 1,
p,qFD(κ,μ;3)[a,b,c,d;e;x,y,z]:= m,n,r=0(a;p,q;κ,μ)m+n+r (b)m (c)n (d)r (e)m+n+r xmm! ynn! zrr!max(|x|, |y|, |z|)< 1,
p,qΦ1(κ,μ)[a,b;d;x,y]:= m,n=0 (a;p,q;κ,μ)m+n (b)m(d)m+n xmm! ynn!max(|x|, |y|)< 1,
p,qΨ1(κ,μ)[a,b;d,e;x,y]:= m,n=0(a;p,q;κ,μ)m+n (b)m(d)m (e)n xmm! ynn!|x|+|y|< 1,
p,qΞ1(κ,μ)[a,c,d;e;x,y]:= m,n=0(a;p,q;κ,μ)m (c)m (d)n(e)m+n xmm! ynn!max(|x|, |y|)< 1
and
p,qΦD(κ,μ;3)[a,b,c;e;x,y,z]:= m,n,r=0(a;p,q;κ,μ)m+n+r (b)m (c)n (e)m+n+r xmm! ynn! zrr!max(|x|, |y|, |z|)< 1,
respectively. Note that taking p = 1, q = 0, κ = 0 and μ = 0 gives the original ones [1, 7, 8, 9, 10, 18, 22, 23, 26, 29, 30, 32, 33]. Now, we obtain the integral representations of the functions (30)(34).

Theorem 11

The following integral representations for(30)hold true:

p,qF1(κ,μ)[a,b,c;d;x,y]= 1Γ(a)0 ta1 exp(tκpqtμ) Φ2[b,c;d;xt,yt]dt
and
p,qF1(κ,μ)[a,b,c;d;x,y]= 1Γ(c)0 tc1 exp(t) p,qΦ1(κ,μ)[a,b;d;x,yt]dt.

Proof

Using the generalization of the extended Pochhammer symbol (a1; p,q;κ,μ) in the definition (30) by its integral representation given by (6), we led to desired result (39). Similar way, we can prove the (40).

Theorem 12

The following integral representations for(31)hold true:

p,qF2(κ,μ)[a,b,c;d,e;x,y]= 1Γ(a)0 ta1 exp(tκpqtμ) 1F1[b;d;xt] 1F1[c;e;yt]dt
and
p,qF2(κ,μ)[a,b,c;d,e;x,y]= 1Γ(c)0 tc1 exp(t) p,qΨ1(κ,μ)[a,b;d,e;x,yt]dt.

Proof

Using the generalization of the extended Pochhammer symbol (a1; p,q;κ, μ) in the definition (31) by its integral representation given by (6), we led to desired result (41). Similar way, we can prove the (42).

Theorem 13

The following integral representation for(32)holds true:

p,qF3(κ,μ)[a,b,c,d;e;x,y]= 1Γ(b)0 tb1 exp(t) p,qΞ1(κ,μ)[a,c,d;e;x,yt]dt.

Proof

Using the generalization of the extended Pochhammer symbol (a1; p,q;κ, μ) in the definition (32) by its integral representation given by (3), we led to desired result (43).

Theorem 14

The following integral representation for(33)holds true:

p,qF4(κ,μ)[a,b;c,d;x,y]= 1Γ(a)0 ta1 exp(tκpqtμ) Ψ2[b;c,d;xt,yt]dt.

Proof

Using the generalization of the extended Pochhammer symbol (a1; p,q;κ, μ) in the definition (33) by its integral representation given by (6), we led to desired result (44).

Theorem 15

The following integral representations for(34)hold true:

p,qFD(κ,μ;3)[a,b,c,d;e;x,y,z]= 1Γ(a)0 ta1 exp(tκpqtμ) Φ2(3)[b,c,d;e;xt,yt,zt]dt
and
p,qFD(κ,μ;3)[a,b,c,d;e;x,y,z]= 1Γ(d)0 td1 exp(t) p,qΦD(κ,μ;3)[a,b,c;e;x,y,zt]dt.

Proof

Using the generalization of the extended Pochhammer symbol (a1; p,q;κ, μ) in the definition (34) by its integral representation given by (6), we led to desired result (45). Similar way, we can prove the (46).

Theorem 16

The following derivative formulas for(30)(34)hold true:

Dx,ym,n{ p,qF1(κ,μ)[a,b,c;d;x,y]}:=(a)m+n(b)m (c)n(d)m+np,qF1(κ,μ)[a+m+n,b+m,c+n;d+m+n;x,y],
Dx,ym,n{ p,qF2(κ,μ)[a,b,c;d,e;x,y]}:=(a)m+n(b)m (c)n(d)m (e)np,qF2(κ,μ)[a+m+n,b+m,c+n;d+m,e+n;x,y],
Dx,ym,n{ p,qF3(κ,μ)[a,b,c,d;e;x,y]}:=(a)m(b)n (c)m (d)n(e)m+np,qF3(κ,μ)[a+m,b+n,c+m,d+n;e+m+n;x,y],
Dx,ym,n{ p,qF4(κ,μ)[a,b;c,d;x,y]}:=(a)m+n(b)m+n(c)m (c)np,qF4(κ,μ)[a+m+n,b+m+n;c+m,d+n;x,y]
and
Dx,y,zm,n,r{ p,qFD(κ,μ;3)[a,b,c,d;e;x,y]}:=(a)m+n+r(b)m (c)n (d)r(e)m+n+r× p,qFD(κ,μ;3)[a+m+n+r,b+m,c+n,d+r;e+m+n+r;x,y,z].

Proof

Differentiating (30)(33) with respect to x and y, then repeating same procedure n-times and making some simple calculation, we can obtain the (47)(50) results. Similiarly, taking differentiation (34) with respect to x, y and z, we can get the derivative formula (51)

Theorem 17

The following derivative formulas for(30)hold true:

Dyn{yc+n1 p,qF1(κ,μ)[a,b,c;d;x,y]}:=(c)n yp,qc1F1(κ,μ)[a,b,c+n;d;x,y],
Dyn{yd1 p,qF1(κ,μ)[a,b,c;d;x,y]}:=(1)n(1d)n ydn1 p,qF1(κ,μ)[a,b,c;dn;x,y]
and
Dyn{ydb1p,qF1(κ,μ)[a,b,c;d;x,y]}:=(1)n(bd1)nydbn1×n=0m(mn)(a)n(c)nyn(d)n(dbm)np,qF1(κ,μ)[a+n,b,c+n;d+n;x,y].

Proof

Multiplying the (30) with yc+n−1 and taking the derivative n-times with respect to y, we have

Dyn{yc+n1 p,qF1(κ,μ)[a,b,c;d;x,y]}:=Dyn{yc+n1 m,n=0(a;p,q;κ,μ)m+n (b)m (c)n(d)m+n xmm! ynn!}=m=0(a)m (b)m xm(d)m m! Dyn{yc+n1 n=0(a+m;p,q;κ,μ)n (c)n(d+m)n n!yn}=m=0(a)m (b)m xm(d)m m! (c)n yc1 n=0(a+m;p,q;κ,μ)n (c+n)n(d+m)n n!yn= (c)n yc1 m,n=0(a;p,q;κ,μ)m+n (b)m (c+n)n(d)m+n xmm! ynn!.
Thus, we obtain the (52) result. Similar way, we can prove the equations (53) and (54).

Theorem 18

The following derivative formulas for(31)hold true:

Dyn{yc+n1 p,qF2(κ,μ)[a,b,c;d,e;x,y]}:=(c)n yp,qc1F2(κ,μ)[a,b,c+n;d,e;x,y]
and
Dyn{ye1 p,qF2(κ,μ)[a,b,c;d,e;x,y]}:=(1)n(1e)n yen1 p,qF2(κ,μ)[a,b,c;d,en;x,y].

Proof

The proof of theorem would be parallel to those of the Theorem 17.

Theorem 19

The following derivative formulas for(32)hold true:

Dyn{yd+n1 p,qF3(κ,μ)[a,b,c,d;e;x,y]}:=(d)n yp,qd1F3(κ,μ)[a,b,c,d+n;e;x,y],
Dyn{(1y)b+n1 p,qF3(κ,μ)[a,b,ec,c;e;x,y]}:=(1)n(b)n (ec)n(e)n (1y)b1× p,qF3(κ,μ)[a,b+n,ec+n,d;e+n;x,y]
and
Dyn{yec1 p,qF3(κ,μ)[a,b,c,d;e;x,y]}:=(1)n(ce1)n yecn1× n=0m(mn) (b)n (d)n yn(e)n (ecm)n p,qF3(κ,μ)[a,b+n,c,d+n;e+n;x,y].

Proof

The proof of theorem would be parallel to those of the Theorem 17.

Theorem 20

The following derivative formulas for(33)hold true:

Dxn{xc1 p,qF4(κ,μ)[a,b;c,d;x,y]}:=(1)n(1c)n xp,qcn1F4(κ,μ)[a,b;cn,d;x,y]
and
Dyn{yd1 p,qF4(κ,μ)[a,b;c,d;x,y]}:=(1)n(1d)n yp,qdn1F4(κ,μ)[a,b;c,dn;x,y].

Proof

The proof of theorem would be parallel to those of the Theorem 17.

5 Recursion Formulas for Extended Appell Hypergeometric Functions

In this section, we present some recursion formulas for Appell hypergeometric functions. Let’s we start following theorem.

Theorem 21

The following recursion formulas for(30)hold true:

p,qF1(κ,μ)[a,b+n,c;d;x,y]=p,qF1(κ,μ)[a,b,c;d;x,y]+ axd k=1n p,qF1(κ,μ)[a+1,b+k,c;d+1;x,y],
p,qF1(κ,μ)[a,bn,c;d;x,y]=p,qF1(κ,μ)[a,b,c;d;x,y] axd k=0n1p,qF1(κ,μ)[a+1,bk,c;d+1;x,y],
and
p,qF1(κ,μ)[a,b,c;dn;x,y]=p,qF1(κ,μ)[a,b,c;d;x,y]+ abx k=1n p,qF1(κ,μ)[a+1,b+1,c;d+2k;x,y](dk)(dk1)+ acy k=1n p,qF1(κ,μ)[a+1,b,c+1;d+2k;x,y](dk)(dk1).

Proof

Applying the transformation formula (b+1)m=(b)m×(1+mb) in the definition of the extension of the Appell hypergeometric function p,qF1(κ,μ)(.) in (30) and we have following contiguous formula:

p,qF1(κ,μ)[a,b+1,c;d;x,y]=p,qF1(κ,μ)[a,b,c;d;x,y]+ axd p,qF1(κ,μ)[a+1,b+1,c;d+1;x,y].
Calculating the function p,qF1(κ,μ)(.) with the parameter b+n by equation (66) for n times, we obtain the required result (63). Setting the b = b − n in the equation (66) and making same calculation as above equation, we can be yield the desired result (64). The proof of (65) is omitted to readers because it is similar to the proof of (63).

Theorem 22

The following recursion formulas for(31)hold true:

p,qF2(κ,μ)[a,b+n,c;d,e;x,y]=p,qF2(κ,μ)[a,b,c;d,e;x,y]+ axd k=1np,qF2(κ,μ)[a+1,b+k,c;d+1,e;x,y],
p,qF2(κ,μ)[a,bn,c;d,e;x,y]=p,qF2(κ,μ)[a,b,c;d,e;x,y] axd k=0n1p,qF2(κ,μ)[a+1,bk,c;d+1,e;x,y],
and
p,qF2(κ,μ)[a,b,c;dn,e;x,y]=p,qF2(κ,μ)[a,b,c;d,e;x,y]+ abx k=1n p,qF2(κ,μ)[a+1,b+1,c;d+2k,e;x,y](dk)(dk1).

Proof

The proof of the Theorem 22 is similar to the proof of Theorem 21.

Theorem 23

The following recursion formula for(32)holds true:

p,qF3(κ,μ)[a,b,c,d;en;x,y]=p,qF3(κ,μ)[a,b,c,d;e;x,y]+ acx k=1n p,qF3(κ,μ)[a+1,b,c+1,d;e+2k;x,y](ek)(ek1)+ bdy k=1n p,qF3(κ,μ)[a,b+1,c,d+1;e+2k;x,y](ek)(ek1).

Proof

The proof of the Theorem 23 is parallel to the proof of Theorem 21.

Theorem 24

The following recursion formula for(33)holds true:

p,qF4(κ,μ)[a,b;cn,d;x,y]=p,qF4(κ,μ)[a,b;c,d;x,y]+ abx k=0n1 p,qF4(κ,μ)[a+1,b+1;c+1k,d;x,y](ck)(ck1).

Proof

The proof of the Theorem 24 is same as the proof of Theorem 21.

Remark 2

Taking p = 1 and q = κ = μ = 0 in the relation Theorem 21Theorem 24, it is easily seen that the special case of recursion formulas of Appell hypergeometric functions [32].

6 Conclusions

We may also give point to that results obtained in this work are of general character and can appropriate to give a new generalization of the Pochhammer symbol by means of the generalization of extended gamma function (4) [31]. Using the generalization of Pochhammer symbol, we give a generalization of the extended hypergeometric functions one or several variables. Also, we obtain various integral representations, derivative formulas and certain properties of these functions.

Acknowledgments

Authors would like to thank reviewers for careful reading of the manuscript and their valuable comments and suggestions for the betterment of the present paper.

References

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If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    M. Abramowitz and I.A. Stegun (eds.). (1965), Handbook of Mathematical Functions with Formulas, Graphs, and-Mathematical Tables, Applied Mathematics Series 55, Tenth Printing, National Bureau of Standards, Washington, DC, 1972; Reprinted by Dover Publications, NewYork.

  • [2]

    P. Agarwal, J. Choi, R. B. Paris, (2015), Extended Riemann-Liouville fractional derivative operator and its applications. Journal of Nonlinear Science and Applications (JNSA) 8 (5).

  • [3]

    P. Agarwal, S.S. Dragomir, M. Jleli, B. Samet, (2019), Advances in Mathematical Inequalities and Applications. Springer, Berlin.

  • [4]

    P. Agarwal, S. Jain, (2011), Further results on fractional calculus of Srivastava polynomials. Bull. Math. Anal. Appl, 3(2), 167–174.

  • [5]

    P. Agarwal, S. Jain, T. Mansour, (2017), Further extended Caputo fractional derivative operator and its applications. Russian Journal of Mathematical Physics, 24(4), 415–425.

  • [6]

    P. Agarwal, J. J. Nieto, (2015), Some fractional integral formulas for the Mittag-Leffler type function with four parameters. Open Mathematics, 13(1).

  • [7]

    G.E. Andrews, R. Askey, R. Roy. (1999), Special Functions, Encyclopedia of Mathematics and Its Applications, Vol. 71, Cambridge University Press, Cambridge, London and New York.

  • [8]

    Y. A. Brychkov and N. Saad. (2012), Some formulas for the Appell function F 1(a,b,b ;c;w,z), Integral Transforms Spec. Funct. 23(11), 793–802.

  • [9]

    M. A. Chaudhry, A. Qadir, M. Raque, S. M. Zubair. (1997), Extension of Euler’s Beta function. J. Compt. Appl. Math. 78: 19–32.

  • [10]

    M. A. Chaudhry, S. M. Zubair. (2002), On a class of incomplete Gamma with Applications. CRC Press (Chapman and Hall), Boca Raton, FL.

  • [11]

    J. Choi, P. Agarwal, (2013), Certain unified integrals associated with Bessel functions. Boundary Value Problems, 2013(1), 95.

  • [12]

    J. Choi, P. Agarwal, (2013), Certain unified integrals involving a product of Bessel functions of first kind. Honam Math. J, 35(4), 667–677.

  • [13]

    J. Choi, P. Agarwal, (2014), Certain fractional integral inequalities involving hypergeometric operators. East Asian Math. J, 30(3), 283–291.

  • [14]

    J. Choi, P. Agarwal, (2014), Certain integral transform and fractional integral formulas for the generalized Gauss hypergeometric functions. In Abstract and Applied Analysis (Vol. 2014). Hindawi.

  • [15]

    J. Choi, P. Agarwal, (2014), Some new Saigo type fractional integral inequalities and their-analogues. In Abstract and Applied Analysis (Vol. 2014). Hindawi.

  • [16]

    J. Choi, P. Agarwal, (2016), A note on fractional integral operator associated with multiindex Mittag-Leffler functions. Filomat, 30(7), 1931–1939.

  • [17]

    J. Choi, A.K. Rathie, R.K. Parmar. (2014), Extension of extended beta, hypergeometric and confluent hypergeometric functions. Honam Mathematical Journal 33: 357–385.

  • [18]

    A. Erdélyi, W. Mangus, F. Oberhettinger, F.G. Tricomi. (1953), Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, Toronto and London.

  • [19]

    İ. O. Kıymaz, A. Çetinkaya, P. Agarwal, (2016), An extension of Caputo fractional derivative operator and its applications. J. Nonlinear Sci. Appl, 9, 3611–3621.

  • [20]

    D. Lee, A. K. Rathie, R. K. Parmar, Y. S. Kim. (2011), Generalization of extended beta function, hypergeometric and confluent hypergeometric functions. Honam Mathematical Journal 33: 187–206.

  • [21]

    M. J. Luo, G. V. Milovanovic, P. Agarwal. (2014), Some results on the extended beta and extended hypergeometric functions. Applied Mathematics Comp. 248: 631–651.

  • [22]

    W. Magnus, F. Oberhettinger, R.P. Soni. (1966), Formulas and Theorems for the Special Functions of Mathematical Physics, Third Enlarged Edition, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtingung der Anwendungsgebiete, vol. 52, Springer-Verlag, Berlin, Heidelberg and New York.

  • [23]

    F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark (eds). (2010), NIST Handbook of Mathematical Functions [With 1 CD-ROM (Windows, Macintosh and UNIX)], U.S. Department of Commerce, National Institute of Standards and Technology, Washington, D.C., 2010; Cambridge University Press, Cambridge, London and New York, (see also AS).

  • [24]

    E. Özergin, M. A. Özarslan, A. Altın. (2011), Extension of Gamma, beta and hypergeometric functions. Journal of Comp. and Applied Math. 235: 4601–4610.

  • [25]

    R. K. Parmar. (2013), A new generalization of Gamma, Beta, hypergeometric and Confluent Hypergeometric functions. Le Mathematiche 68: 33–52.

  • [26]

    E.D. Rainville. (1971), Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea publishing Company, Bronx, New York.

  • [27]

    M. Ruzhansky, Y.J. Cho, P. Agarwal, I. Area, (2017), Advances in Real and Complex Analysis with Applications. Springer, Berlin.

  • [28]

    H. M. Srivastava, P. Agarwal, S. Jain, (2014), Generating functions for the generalized Gauss hypergeometric functions. Applied Mathematics and Computation, 247, 348–352.

  • [29]

    H. M. Srivastava, A. Çetinkaya, İ. O. Kıymaz. (2014), A certain generalized Pochhammer symbol and its applications to hypergeometric functions. Applied Mathematics and Computation 226: 484–491.

  • [30]

    H.M. Srivastava, H.L. Manocha. (1984), A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto.

  • [31]

    R. Şahin, O. Yağcı, M. B. Yağbasan, A. Çetinkaya, İ. O. Kıymaz. (2018), Further Generalizations of Gamma, Beta and Related Functions. Journal Of Inequalities And Special Functions 9: 1–7.

  • [32]

    X. Wang. (2012), Recursion formulas for Appell functions, Integral Transforms Spec. Funct. 23(6), 421–433.

  • [33]

    G.N. Watson. (1944), A Treatise on the Theory of Bessel Functions, Second edition, Cambridge University Press, Cambridge, London and New York.