Some Generalized Special Functions and their Properties

Available online at www.atnaa.org Research Article

Some Generalized Special Functions and their Properties

Shahid Mubeen^a, Syed Ali Haider Shah^a, Gauhar Rahman^b, Kottakkaran Sooppy Nisar^c, Thabet Abdeljawad^d

aDepartment of Mathematics, University of Sargodha, Sargodha, Pakistan.

bDepartment of Mathematics, Shaheed Benazir Bhutto University, Sheringal, Upper Dir, Pakistan.

cDepartment of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser, Saudi Arabia.

dDepartment of Mathematics, Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia.

Abstract

In the present paper, rst, we investigate a generalized Pochhammer's symbol and its various properties in terms of a new symbol (s, k), where s, k > 0. Then, we dene a generalization of gamma and beta functions and their various associated properties in the form of (s, k). Also, we dene new generalization of hyperge- ometric functions and develop dierential equations for generalized hypergeometric functions in the form of (s, k). We present that generalized hypergeometric functions are the solution of the said dierential equa- tion. Furthermore, some useful results, properties and integral representation related to these generalized Pochhammer's symbol, gamma function, beta function, and hypergeometric functions are presented.

Keywords: Pochhammer symbol beta function Gamma function generalized Pochhammer symbol generalized hypergeometric function.

2010 MSC: 33C20, 33C05, 33C10; 26A09.

1. Introduction

The theory of special functions comprises a major part of mathematics. In last three centuries, the essential of solving the problems take place in the elds of classical mechanics, hydrodynamics and control

Email addresses: smjhanda@gmail.com (Shahid Mubeen), ali.bukhari78699@gmail.com (Syed Ali Haider Shah), gauhar55uom@gmail.com (Gauhar Rahman), n.sooppy@psau.edu.sa, ksnisar1@gmail.com (Kottakkaran Sooppy Nisar), tabdeljawad@psu.edu.sa (Thabet Abdeljawad)

Received July 14, 2020, Accepted October 09, 2021, Online October 16, 2021

theory, motivated the development of the theory of special functions. This eld also has wide applications in both pure mathematics and applied mathematics. Numerous extensions of special functions have been introduced by many authors (see [1, 2, 3, 4]).

Agarwal et al. [5] proved new dierential equations for the extended Mittag-Leer function by using Saigo-Maeda fractional dierential operators. Mdallal et al. [6] also worked on the dierential equations of fractional order with variables coecients. They found the eigenvalues by applying associated boundary conditions. Also, they present the eigenfunctions in the form of Mittag-Leer functions. Babakhani et al.

[7] used thye xed point theorems to nd the existence of positive solutions for a non-autonomous fractional dierential equation with integral boundary conditions. They also presented some examples related to dierential equations. Jarad et al. [8] investigated the modied Laplace transform and related properties.

They used these results to nd the solution of some ordinary dierential equations of a certain type generalized fractional derivatives.

Diaz et al. [9, 10, 11] have introduced gamma and beta k-functions and proved a number of their properties. They have also studied zeta k-functions and hypergeometric k-functions based on Pochhammer k-symbols for factorial functions. In [12, 13, 14, 15], the researchers studied the generalized gamma k- function and proposed its various properties. Later on, Mubeen and Habibullah [32] proposed the so-called k-fractional integral based on gamma k-function and its applications. In [33], Mubeen and Habibullah dened the integral representation of generalized conuent hypergeometric k-functions and hypergeometric k-functions by utilizing the properties of Pochhammer k-symbols, k-gamma, and k-beta functions. In [34], Mubeen et al. proposed the following second order linear dierential equation for hypergergeometric k- functions as

kω(1 − kx)ω⁰⁰+ [γ − (α + β + k) kx] ω⁰− αβω = 0.

The solution in the form of the so-called k-hypergeometric series of k-hypergeometric dierential equation by utilizing the Frobenius method can be found in the work of Mubeen et al. [35, 36]. Recently, Li and Dong [37] investigated the hypergeometric series solutions for the second order non-homogeneous k-hypergeometric dierential equation with the polynomial term. Rahman et al. [38, 39] proposed the generalization of Wright type hypergeometric k-functions and derived its various basic properties.

Qi and Wang [40] worked on Young's integral Inequalities and discussed its geometric interpretation. Ad- jimi and Benbachir [41] worked on Katugampola fractional dierential equation with Erdelyi-Kober integral boundary conditions. Furthermore, Mubeen and Iqbal [16] investigated the generalized version of Grüss-type inequalities by considering k-fractional integrals. Agarwal et al. [18] established certain Hermite-Hadamard type inequalities involving k-fractional integrals. Set et al. [27] proposed generalized Hermite–Hadamard type inequalities for Riemann–Liouville k-fractional integral. Östrowski type k-fractional integral inequali- ties can be found in the work of Mubeen et al. [17]. Many researchers have established further the generalized versions of Riemann-Liouville k-fractional integrals, and dened a large number of various inequalities by using dierent kind of generalized fractional integrals. The interesting readers may consult [19, 31, 25, 20].

The Hadamard k-fractional integrals can be found in the work of Farid et al. [21]. In [23] Farid proposed the idea of Hadamard-type inequalities for k-fractional Riemann-Liouville integrals. In [24, 22], the authors have introduced the inequalities by employing Hadamard-type inequalities for k-fractional integrals. Nisar et al. [28] investigated Gronwall type inequalities by utilizing Riemann-Liouville k-fractional derivatives and Hadamard k-fractional derivatives [28]. In [28], they presented dependence solutions of certain k-fractional dierential equations of arbitrary real order with initial conditions. Samraiz et al. [26] proposed Hadamard k-fractional derivative and related properties. Recently, Rahman et al. [29] dened generalized k-fractional derivative operator. Jangid et al. [30] worked on the generalization of integral inequalities. By using the Saigo k-fractional integral operators, they found some new inequalities for the Chebyshev functional for two synchronous functions.

By getting motivation from Diaz and other researchers working on the generalization and extensions of the special functions, we get the idea to obtain the more generalized and extended form of special functions.

The structure of the paper is organized as follows:

In Section 2, we study the generalized Pochhammer's symbol, gamma function and beta function in terms of (s, k). Also, some basic properties are presented. In Section 3, generalized hypergeometric functions, dierential equation for hypergeometric function and their associated properties are discussed. In Section 4, concluding remarks are given.

2. Main Results

In this section, we introduce generalized Pochhammer's symbol and its various associated properties.

Also, generalization of gamma and beta functions and their associated properties are presented.

Denition 2.1. The generalized Pochhammer's symbol in term of (s, k) is dened as

s(α)^k_n= α(α + (s/k)) · · · (α + ((n − 1)s/k)), α ∈ C,

s, k ∈ R⁺, n ∈ N, s > k > 0. (1)

s(b)^k_nm= 2^{mn s}(_m^b)^k_n ^s(^bk+s_mk )^k_n· · · ^s(^bk+(m−1)s_mk )^k_n

Just as for the usual Γ, the function^sΓ^k admits an innite product expression given by 1

sΓ^k(z) = z(s/k)^−(kz)/se^(kzγ)/s

∞

Y

n=1



1 + kz ns

 exp



−kz ns



, (2)

where γ is Euler-Mascheroni constant, dened by γ = lim

n→∞(H_n− log(n)) ≈ 0.57721566490, where Hn=

Pn m=1

1 m.

Lemma 2.2. For all z, the Euler product is given as

sΓ^k(z) = 1 z(s/k)^−kzs

∞

Y

n=1

 n + 1 n

^kz_s  1 +kz

sn

−1

. (3)

Proof. Using equation (2), we have 1

sΓ^k(z) = z(s/k)^−−kzs e^kγzs

∞

Y

n=1



1 +kz ns



exp −kz ns



.

z(s/k)^{−kzs s}Γ^k(z) = e^−kγzs lim

n→∞

n

Y

m=1



1 + kz sm

−1

exp kz sm



. (4)

Since

γ = lim

n→∞H_n− log(n + 1). (5)

Therefore equation (5) becomes

γ = lim

n→∞

 H_n−

n

X

m=1

log m + 1 m



. (6)

Multiply the whole equation (6) by −^kz_s , we obtain

−kzγ

s = lim

n→∞



− kz s H_n+

n

X

m=1

log m + 1 m

^kz_s  . By taking exponential, we get

e^−kzγs = lim

n→∞

 exp



−kz s H_n

 exp

n

X

m=1

log m + 1 m

^kz_s 

. (7)

Since Hn=

n

P

m=1 1

m, put in equation (7), we have e^−kzγs = lim

n→∞

 exp



−kz s

n

X

m=1

1 m

 exp

n

X

m=1

log m + 1 m

^kz_s 

= lim

n→∞

 exp

 n

X

m=1

−kz sm

 exp

n

X

m=1

log m + 1 m

^kz

s  . And so

e^−kzγs = lim

n→∞exp

 n

X

m=1

log



exp −kz sm



exp

n

X

m=1

log m + 1 m

^kz_s



e^−kzγs = lim

n→∞

n

Y

m=1

exp −kz sm

 m + 1 m

^kz_s

. (8)

This implies that

sΓ^k(z) = 1 z(s/k)^−kzs

∞

Y

n=1

 n + 1 n

^kz_s

1 +kz sn

−1 .

Lemma 2.3. For all nite z, the dierence equation is given below

sΓ^k(z + (s/k)) = z ^sΓ^k(z). (9)

Proof. Using equation (3), we have

sΓ^k(z + (s/k))

sΓ^k(z) =

1

(z+(s/k))(s/k)^{−k(z+(s/k))s}

∞

Q

n=1

[(ⁿ⁺¹_n )^k(z+(s/k))s (1 +^kz+k_sn )⁻¹]

1 z(s/k)^{− kzs}

∞

Q

n=1

[(ⁿ⁺¹_n )^kzs (1 +^kz_ns)⁻¹]

= z(s/k) z + (s/k) lim

n→∞

n

Y

m=1

(m + 1

m )( sm + kz sm + kz + k)

= z(s/k) z + (s/k) lim

n→∞

 (n + 1

1 )( s + zk sn + kz + k)



= z.

Therefore

sΓ^k(z + (s/k)) = z ^sΓ^k(z).

Denition 2.4. The Gamma (s, k)-function is dened as

sΓ^k(z) = lim

n→∞

n!sⁿ(^ns_k)^kzs k^{n s}(z)^k_n

, z ∈ C − kZ, (10)

s, k ∈ R⁺, s > k > 0, <(z) > 0. (11) Lemma 2.5. For z ∈ C, <(z) > 0, we have

sΓ^k(z) =

∞

Z

0

t^z−1e^−kts/ks dt.

Proof. By equation (10), we have

sΓ^k(z) =

∞

Z

0

t^z−1e^−kts/ks dt

= lim

n→∞

(^ns_k)^ks

Z

0

t^z−1



1 −kt^sk ns

 dt.

Let An,i(z), i = 0, · · · , n, be given by

A_n,i(z) =

(^ns_k)^ks

Z

0

t^z−1



1 −kt^sk ns

 dt.

The following recursive formula is proven using integration by parts An,i(z) = i

nzAn,i(z + s k).

Also,

A_n,0(z) =

(^ns_k)^ks

Z

0

t^z−1dt = (^ns_k)^kzs z . Therefore,

A_n,n(z) = n!sⁿ(^ns_k)^kzs k^{n s}(z)^k_n(1 + ^zk_ns) and

sΓ^k(z) = lim

n→∞A_n,n(z) = lim

n→∞

n!sⁿ(^ns_k)^kzs k^{n s}(z)^k_n .

sΓ^k(z) =

∞

Z

0

t^z−1e^−kts/ks dt, z ∈ C, s, k ∈ R⁺, s > k > 0, <(z) > 0. (12)

Lemma 2.6. If α is neither zero nor a negative integer, then

s(α)^k_n=

sΓ^k α +^ns_k

sΓ^k(α) . (13)

Proof. Consider

sΓ^k

 α +ns

k



=^sΓ^k



α +(n − 1 + 1)s k

 . Using equation (9), for n a positive integer, we have

sΓ^k

 α + ns

k



=



α +ns − s k



sΓ^k



α +ns − s k



=



α +ns − s k



α +ns − 2s k



· · · α ^sΓ^k(α) i.e.,

sΓ^k

 α +ns

k



= α

 α + s

k



· · ·



α + ns − 2s k



α +ns − s k



sΓ^k(α).

Since

s(α)^k_n = α

 α + s

k



· · ·



α +ns − 2s k



α +ns − s k

 , therefore we have

sΓ^k

 α + ns

k



= ^s(α)^k_n ^sΓ^k(α).

This implies that

s(α)^k_n=

sΓ^k α +^ns_k

sΓ^k(α) .

Denition 2.7. The beta (s, k)-function is dened as

sβ^k(x, y) = k s

1

Z

0

t^kxs−1(1 − t)^{kys −1}dt, s, k ∈ R⁺, s > k > 0, <(x) > 0, <(y) > 0. (14)

The beta (s, k)-function can be dened in terms of algebraic functions by putting t = sin²Φ, given as

sβ^k(x, y) = 2k s

π

Z2

0

sin^{2kxs −1}Φ cos^{2kys −1}ΦdΦ, <(x) > 0, <(y) > 0. (15)

Lemma 2.8. If Re(p) > 0 and Re(q) > 0, then

sβ^k(p, q) =

sΓ^k(p)^sΓ^k(q)

sΓ^k(p + q) . (16)

Proof. Using equation (12), We have

sΓ^k(p)^sΓ^k(q) =

∞

Z

0

e^−ku

s k

s u^kps−1du

∞

Z

0

e^−kv

s k

s v^kqs−1dv. (17)

In equation (17), use u = x² and v = y². Thus du = 2xdx and dv = 2ydy.

Now if u → 0, then x → 0. If v → 0, then y → 0. If u → ∞, then x → ∞. If v → ∞, then y → ∞. We have

sΓ^k(p) ^sΓ^k(q) =

∞

Z

0

e^−kx

2s k

s x^{2kps −2}2xdx

∞

Z

0

e^−ky

2s k

s y^{2kqs −2}2ydy

= 4

∞

Z

0

∞

Z

0

e^−(kx

2s k s +^ky

2s k

s )x^{2kps −1}y^{2kqs −1}dxdy. (18)

Let x = r cos θ, y = r sin θ and r² = x²+ y², where 0 < r < ∞ and 0 < θ < ^π₂, and dxdy = rdrdθ. Put in the equation (18), we have

sΓ^k(p) ^sΓ^k(q) = 4

∞

Z

0

π

Z2

0

e^−kr

2s k

s r^2p−1cos^2kps−1θr^2q−1sin^{2kqs −1}θrdrdθ

= 2

∞

Z

0

e^−kr

2s k

s r^2p+2q−22rdr

π

Z2

0

cos^{2pks −1}θ sin^{2kqs −1}θdθ. (19)

Put the following in equation (19)

r² = t θ = π

2 − φ

2rdr = dt dθ = −dφ,

as r → 0, t → 0 and as r → ∞, t → ∞. Also if θ = 0, then φ = ^π₂ and if θ = ^π₂, then φ = 0, so we have

sΓ^k(p) ^sΓ^k(q) =

∞

Z

0

e^−kt

s k

s t^p+q−1dt 2

π 2

Z

0

sin^{2kps −1}φ cos^{2kqs −1}φdφ

= ^sΓ^k(p + q)^sβ^k(p, q).

By using equation (12) and equation (15) we have

sβ^k(p, q) =

sΓ^k(p)^sΓ^k(q)

sΓ^k(p + q) .

Lemma 2.9. Prove that

(1 −sy

k)^−kas =

∞

X

n=0

s(a)^k_n yⁿ

n! . (20)

Proof. The binomial theorem states that

(1 − sy

k )^−kas =

∞

X

n=0

(−^ka_s)(−^ka_s − 1)(−^ka_s − 2)...(−^ka_s − n + 1)(−1)ⁿ(^sy_k)ⁿ

n! ,

which may be written as

(1 −sy

k )^−kas =

∞

X

n=0

(a)(a +_k^s)(a + 2_k^s)...(a + (n − 1)_k^s) yⁿ

n! .

Therefore, in factorial function notation,

(1 −sy

k )^−kas =

∞

X

n=0

s(a)^k_nyⁿ n! .

2.1. Some important results of Pochhammer (s, k)-symbol.

In this section, some important results of Pochhammer's (s, k)-symbol are presented.

sΓ^k(^s−ns_k − α)

sΓ^k(_k^s − α)

=

sΓ^k(^s−ns_k − α)

(^s−s_k − α)(^s−2s_k − α) · · · (^s−ns_k − α) ^sΓ^k(^s−ns_k − α)

= (−1)ⁿ

α(α + ^s_k) · · · (α + (n − 1)^s_k) = (−1)ⁿ

s(α)^k_n. Thus,

sΓ^k(^s−ns_k − α)

sΓ^k(^s_k− α) = (−1)ⁿ

s(α)^k_n.

s(α)^k_m ^s(α + ms

k )^k_n= α(α + s

k) · · · (α + (m − 1)s

k )(α + ms

k )(α +(m + 1)s k ) · · · (α +ms + (n − 1)s

k ) =^s(α)^k_n+m. Thus,

s(α)^k_m ^s(α + ms

k )^k_n=^s (α)^k_n+m. (−1)^{m s}(α)^k_n

s(^s−ns_k − α)^k_m

= (−1)^mα(α + ^s_k) · · · (α + ^(n−1)s_k ) (^s−ns_k − α)(^2s−ns_k − α) · · · (s−nk+(m−1)s

k − α)

= α(α + s

k) · · · (α +(n − m − 1)s

k ) =^s(α)^k_n−m. Thus,

(−1)^{m s}(α)^k_n

s(^s−ns_k − α)^k_m =^s(α)^k_n−m. Example 2.10. Prove that

(n − m)! = (−s)^mn!

k^{m s}(−^ns_k )^k_m.

Proof.

s(α)^k_n−m= (−1)^{m s}(α)^k_n

s(^s−ns_k − α)^k_m, taking α = _k^s, we have

s(s

k)^k_n−m= (−1)^{m s}(_k^s)^k_n

s(^−ns_k )^k_m ,

(s

k)^n−m(n − m)! = (−1)^m (^s_k)ⁿn!

s(^−ns_k )^k_m ,

(n − m)! = (−s)^mn!

k^{m s}(−^ns_k)^k_m.

2.1.1. Legender's (s, k)-Duplication formula.

Lemma 2.11. Prove that

rkπ s

sΓ^k(2z) = 2^{2kzs −1 s}Γ^k(z) ^kΓ^k(z + s 2k).

Proof. Since^s(α)^k_2n= 2^{2n s}(^α₂)^k_n ^s(^kα+s_2k )^k_n and α = 2z, we have

s(2z)^k_2n= 2^{2n s}(2z

2 )^k_n ^s(2kz + s 2k )^k_n

sΓ^k(2z + ^2ns_k )

sΓ^k(2z) = 2^{2n s}Γ^k(^2z₂ + ^ns_k) ^sΓ^k(^2kz+s+2ns_2k )

sΓ^k(^2z₂) ^sΓ^k(^2kz+s_2k ) . Using

sΓ^k(z) = lim

n→∞

n!sⁿ(^ns_k)^kzk k^{n s}(z)^k_n and

1 = lim

n→∞

n!sⁿ(^ns_k)^kzk k^{n s}Γ^k(z +^ns_k).

sΓ^k(2z)

sΓ^k(z)^sΓ^k(z +_2k^s ) = lim

n→∞

sΓ^k(2z +^2ns_k )n!k²ⁿ sⁿ(^ns_k)^kzs (2n)!s²ⁿkⁿ(^2ns_k )^{2kzs s}Γ^k(z + ^ns_k)

× n!sⁿ(^ns_k)^kzs (2n)!s²ⁿ(^2ns_k )^2kzs k²ⁿ (^ns_k )^4kz−3ss k²ⁿkⁿn!n!2^{2n s}Γ^k(z +^s+2ns_2k )

=r s k lim

n→∞

(2n)!√ n 2²ⁿ(n!)² =r s

kC.

Put z = _2k^s in above equation and use^sΓ^k(_2k^s) = qkπ

s . This implies that C =

r1 π. Therefore

rkπ s

sΓ^k(2z) = 2^{2kzs −1 s}Γ^k(z) ^kΓ^k(z + s 2k).

Denition 2.12. The binomial (s, k)-theorem states that

(1 + sz/k)ⁿ=n 0



(sz/k)⁰+n 1



(sz/k)¹+ · · ·n n



(sz/k)ⁿ+ · · · , (21) i.e.

(1 + sz/k)ⁿ=

∞

X

m=0

 n m



(sz/k)^m (22)

and

(1 + sz/k)ⁿ= 1 + (nsz/k) +n(n − 1)(sz/k)²

2! + · · · . (23)

Also

(1 − sz/k)^−kαs =

∞

X

n=0

s(α)^k_nzⁿ

n! , α ∈ R, s > k > 0. (24)

3. Generalized hypergeometric functions and their properties

In this section, a new generalization of hypergeometric functions are introduced with the help of new generalized Pochhammer's symbol (1). We also introduce the generalized hypergeometric dierential equa- tions and derive that the new generalized hypergeometric function is the solutions of the said dierential equations. Certain basic properties are also presented.

Theorem 3.1. Prove that sz(k − sz)ω⁰⁰+ [k²c − (ak + bk + s)sz]ω⁰− abk²ω = 0 if and only if ^s₂F₁^k(a; s, k), (b; s, k)

(c; s, k)     z



=

∞

P

n=0

s(a)^k_n ^s(b)^k_nzⁿ

s(c)^k_nn! is a solution.

Proof. Let ω =^s₂F₁^k(a; s, k), (b; s, k) (c; s, k)

    z



be a solution of the dierential equation. Now Consider θ = z_dz^d

ω =

∞

X

n=0

s(a)^k_n ^s(b)^k_nzⁿ

s(c)^k_nn! . (25)

s kθ s

kθ + c − s k



ω = s

kθ s

kθ + c −s k

 ∞

X

n=0

s(a)^k_n ^s(b)^k_nzⁿ

s(c)^k_n n!

= sz kθ

 a + sθ

k



b + sθ k

 ω.

Thus s kθ s

kθ + c − s k

 ω − sz

kθ

 a +sθ

k



b +sθ k



ω = 0, (26)

after a little simplication, we get

sz(k − sz)ω⁰⁰+ [k²c − (ak + bk + s)sz]ω⁰− abk²ω = 0.

Conversely:

Let sz(k − sz)ω⁰⁰+ [k²c − (ak + bk + s)sz]ω⁰− abk²ω = 0, suppose that

ω =

∞

X

n=0

dnzⁿ

ω⁰ =

∞

X

n=1

nd_nzⁿ⁻¹

ω⁰⁰ =

∞

X

n=2

n(n − 1)dnzⁿ⁻², put these values in dierential equation

sz(k − sz)

∞

X

n=2

n(n − 2)d_nzⁿ⁻¹+ [k²c − (ak + bk + s)sz]

∞

X

n=1

nd_nzⁿ⁻¹− abk²

∞

X

n=0

d_nzⁿ= 0, therefore

sk

∞

X

n=2

n(n − 1)d_nzⁿ⁻¹− s

∞

X

n=2

n(n − 1)d_nzⁿ+ k²c

∞

X

n=1

nd_nzⁿ⁻¹

− [abk²+ (ak + bk + s)s]

∞

X

n=0

d_nzⁿ= 0 replace n by (n + 1)

sk

∞

X

n=1

n(n + 1)d_n+1zⁿ− s

∞

X

n=2

n(n − 1)d_nzⁿ+ k²c

∞

X

n=0

(n + 1)d_n+1zⁿ

− [abk²+ (ak + bk + s)s]

∞

X

n=0

d_nzⁿ= 0.

comparing the coecient of zⁿ on both sides

skn(n + 1)dn+1− sn(n − 1)d_n+ k²c(n + 1)dn+1− [abk²+ (ak + bk + s)s]dn= 0.

Therefore

d_n+1 = sn(n − 1) + [abk²+ (ak + bk + s)s]

skn(n + 1) + k²c(n + 1) d_n

= (a + (ns/k))(b + (ns/k)) (n + 1)(c + (sn/k)) dn. This implies

dn=

s(a)^k_n ^s(b)^k_n

s(c)^k_nn!

ω =

∞

X

n=0

s(a)^k_n^s(b)^k_nzⁿ

s(c)^k_nn!

ω = ^s₂F₁^k(a; s, k), (b; s, k) (c; s, k)

    z



s

2F₁^k(a; s, k), (b; s, k) (c; s, k)

    z



=

∞

X

n=0

s(a)^k_n ^s(b)^k_nzⁿ

s(c)^k_nn! .

Denition 3.2. The hypergeometric (s, k)-function is dened as

s

2F₁^k(a; s, k), (b; s, k) (c; s, k)

    z



=

∞

X

n=0

s(a)^k_n ^s(b)^k_nzⁿ

s(c)^k_nn! , (27)

provided a, b, c ∈ C, s > k > 0, |z| < ^k_s and <(c − b − a) > 0.

Theorem 3.3. If <(c) > <(b) > 0, s > k > 0, then for all nite z < ^k_s

s

2F₁^k(a; s, k), (b; s, k) (c; s, k)

    z



= k ^sΓ^k(c) s ^sΓ^k(b) ^sΓ^k(c − b)

1

Z

0

t^kbs−1(1 − t)^{kc−kbs −1}



1 −sxt k

−^ka

s

dt.

Proof.

s

2F₁^k(a; s, k), (b; s, k) (c; s, k)

    z



=

∞

X

n=0

s(a)^k_n ^s(b)^k_n zⁿ

s(c)^k_n n!

=

∞

X

n=0

s(a)^k_n ^s(b)^k_n zⁿ

s(c)^k_nn! , by using^s(a)^k_n= ^{sΓk(a+(ns)/k)}_sΓk(a) , we have

s

2F₁^k(a; s, k), (b; s, k) (c; s, k)

    z



=

∞

X

n=0

s(a)^k_n ^sΓ^k(b + (ns)/k) ^sΓ^k(c) zⁿ

sΓ^k(c + (ns)/k) ^sΓ^k(b) n!

=

sΓ^k(c)

sΓ^k(b)^sΓ^k(c − b)

∞

X

n=0

s(a)^k_n ^sΓ^k(b + (ns)/k) ^sΓ^k(c − b) zⁿ

sΓ^k(c + (ns)/k) n!

= k^sΓ^k(c) s^sΓ^k(b)^sΓ^k(c − b)

1

Z

0

t^kbs−1(1 − t)^{kc−kbs −1}



1 −szt k

−^ka

s

dt.

Theorem 3.4. If <(c − b − a) > 0, s > k > 0, then

s

2F₁^k(a; s, k), (b; s, k) (c; s, k)

    k s



=

sΓ^k(c)^sΓ^k(c − b − a)

sΓ^k(c − b)^sΓ^k(c − a).

Proof. Let |z| < ^k_s and consider

s

2F₁^k(a; s, k), (b; s, k) (c; s, k)

    k s



= k ^sΓ^k(c) s^sΓ^k(b)^sΓ^k(c − b)

1

Z

0

t^kbs−1(1 − t)^{kc−kbs −1} 1 − t
−^ka

s dt

=

sΓ^k(c)^sΓ^k(c − b − a)

sΓ^k(c − b)^sΓ^k(c − a) .

Theorem 3.5. If |z| < ^k_s and |(1−(s/k)z)|^|z| < 1, s > k > 0, then

s

2F₁^k(a; s, k), (b; s, k) (c; s, k) z



= (1 − (s/k)z)^{−kas s}₂F₁^k(a; s, k), (c − b; s, k) (c; s, k)

−kz k − sz

 . Proof. Let |z| < ^k_s and consider

(1 − (s/k)z)^{−kas s}₂F₁^k(a; s, k), (c − b; s, k) (c; s, k)

−kz k − sz



=

∞

X

m=0

s(a)^k_m ^s(c − b)^k_m z^m(1 − (s/k)z)^{−kas −m}

s(c)^k_m m!

=

∞

X

n=0

s(a)^k_n ^s(b)^k_n zⁿ

s(c)^k_nn!

=^s₂F₁^k(a; s, k), (b; s, k) (c; s, k) z

 .

Theorem 3.6. If |z| < ^k_s, s > k > 0, then

s

2F₁^k(a; s, k), (b; s, k) (c; s, k)

    z



= (1 − (s/k)z)^{−kc−ka−kbs s}₂F₁^k(c − a; s, k), (c − b; s, k) (c; s, k)

    z

 . Proof. Let |z| < ^k_s and consider

s

2F₁^k(a; s, k), (b; s, k) (c; s, k)

    z



= (1 − (s/k)z)^{−kas s}₂F₁^k(a; s, k), (c − b; s, k) (c; s, k)

−kz k − sz

 . Suppose that y = _k−sz^−kz, we have

(1 − (s/k)z)^{kas s}₂F₁^k(a; s, k), (b; s, k) (c; s, k) z



=^s₂F₁^k(a; s, k), (c − b; s, k) (c; s, k)

    y



= (1 − (s/k)y)^{kb−kcs s}₂F₁^k(c − b; s, k), (c − a; s, k) (c; s, k)

−ky k − sy

 . This implies that

s

2F₁^k(a; s, k), (b; s, k) (c; s, k) z



= (1 − (s/k)z)^{−kc−ka−kbs s}₂F₁^k(c − a; s, k), (c − b; s, k) (c; s, k)

    z

 .

Lemma 3.7. Show that

s

2F₁^k(^−ns_k ; s, k), (_k^s − b −^ns_k; s, k) (a; s, k)

    k s



=

s(a + b − ^s_k)^k_n

s(a)^k_n ^s(a + b − _k^s)^k_n. Proof. Since Gauss summation (s, k)-theorem

s

2F₁^k(^−ns_k ; s, k), (^s_k− b −^ns_k; s, k) (a; s, k)

    k s



=

sΓ^k(a)^sΓ^k(a + b + ^2ns−s_k )

sΓ^k(a + ^sn_k) ^sΓ^k(a + b + ^ns−s_k )

=

s(a + b − ^s_k)^k

s(a)^{k s}(a + b − _k^s)^k.

Theorem 3.8. If 2b is neither zero nor a negative integer and if |sy| < ^k₂ and _k−sy^sy 

< 1, s > k > 0, then k^kas (k − sy)^{−kas s}₂F₁^k(^a₂; s, k), (^ka+s_2k ; s, k)

(_2k^s + b; s, k)    

sky² (k − sy)²



=^s₂F₁^k(a; s, k), (b; s, k) (2b; s, k)

2y

 . Proof. Let |sy| < ^k₂ and _k−sy^sy

< 1, we have

k^kas (k − sy)^{−kas s}₂F₁^k(^a₂; s, k), (^ka+s_2k ; s, k) (_2k^s + b; s, k)

sky² (k − sy)²



=

∞

X

n=0

∞

X

m=0

s(^a₂)^k_m ^s(^ka+s_2k )^k_m ^s(a + ^2ms_k )^k_n y^n+2ms^m k^{m s}(b +_2k^s)^k_m n!m!

=^s₂F₁^k(a; s, k), (b; s, k) (2b; s, k)

2y

 .

Denition 3.9. Conuent hypergeometric (s, k)-function is a solution of conuent hypergeometric dierential (s, k)-equation which is a degenerate form of a hypergeometric dierential (s, k)-equation where two regular singularities out of three merge into an irregular singularity.

We dene the conuent hypergeometric (s, k)-function as

s

1F₁^k(a; s, k) (c; s, k)     z



=

∞

X

n=0

s(a)^k_n zⁿ

s(c)^k_n n!, (28)

for |z| < ∞, c 6= 0, −s/k, −2s/k, · · · .

Theorem 3.10. If <(b) > <(a) > 0, s > k > 0, then for all nite z

s

1F₁^k(a; s, k) (b; s, k)     z



= k ^sΓ^k(b) s^sΓ^k(a) ^sΓ^k(b − a)

1

Z

0

t^kas−1(1 − t)^{kb−kas −1}e^xtdt.

Proof.

s

1F₁^k(a; s, k) (b; s, k)     z



=

∞

X

n=0

s(a)^k_n zⁿ

s(b)^k_n n!. By using^s(a)^k_n= ^{sΓk(a+(ns)/k)}sΓ^k(a) , we have

s

1F₁^k(a; s, k) (b; s, k)     z



= k^sΓ^k(b) s^sΓ^k(a) ^sΓ^k(b − a)

1

Z

0

t^kas−1(1 − t)^{kb−kas −1}e^ztdt.

Denition 3.11. The Generalized hypergeometric (s, k)-functions is dened as

srF_q^k(α₁; s, k), (α₂; s, k), · · · , (α_r; s, k) (β₁; s, k), (β₂; s, k), · · · , (β_q; s, k)     z



=

∞

X

n=0

s(α₁)^k_n ^s(α₂)^k_i · · · ^s(α_r)^k_n zⁿ

s(β₁)^k_n ^s(β₂)^k_n · · · ^s(β_q)^k_nn! , (29) for all αi, βj ∈ C, βj 6= 0, −s/k, −2s/k, · · · , |z| < ^k_s where i = 1, 2, · · · , r and j = 1, 2, · · · , q.

It is known that

(i) if r ≤ q, the series converges for all nite z,

(ii) if r = q + 1, the series converges for |z| < ^k_s and diverges for |z| > ^k_s, (iii) if r > q + 1, the series diverges for z 6= 0 unless the series terminates.

Theorem 3.12. If r ≤ s + 1, if <(b1) > <(a₁) > 0, if no one of b1, b₂, · · · b_s is zero or a negative integer, s > k > 0and if |z|^k_s, then

srF_q^k(a₁; s, k), (a₂; s, k), · · · , (a_r; s, k) (b₁; s, k), (b₂; s, k), · · · , (b_q; s, k)

    z



= k ^sΓ^k(b₁) s ^sΓ^k(a₁)^sΓ^k(b₁− a₁)

×

1

Z

0

t^{ka1s −1}(1 − t)^{kb1−ka1s −1} _r−1^sF_q−1^k (a₂; s, k), · · · , (a_r; s, k) (b₂; s, k), · · · , (b_q; s, k)    

zt

 dt.

Proof.

srF_q^k(a₁; s, k), (a₂; s, k), · · · , (a_r; s, k) (b₁; s, k), (b₂; s, k), · · · , (b_q; s, k)     z



=

∞

X

n=0

s(a1)^k_n ^s(a2)^k_n· · ·^s(ar)^k_n zⁿ

s(b1)^k_n ^s(b2)^k_n· · ·^s(bq)^k_n n!

= k ^sΓ^k(b1) s^sΓ^k(a₁)^sΓ^k(b₁− a₁)

×

1

Z

0

t^{ka1s −1}(1 − t)^{kb1−ka1s −1} _r−1^sF_q−1^k (a₂; s, k), · · · , (a_r; s, k) (b₂; s, k), · · · , (b_q; s, k)    

zt

 dt.