On a Basic Analogue of Generalized Hfunction with the help of fractional q-integral operator of Kober type

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On a Basic Analogue of Generalized

H-function with the help of fractional q-integral

operator of Kober type

Naseer Ahmad Malik

Central University of Jharkhand

Brambe, Ranchi-835205 India.

Abstract. In this paper, our objective is to investigate the basic analogue of a new hypergeometric function, which is a generalization of the basic I-function. In this regard, the application of Kober type q-integral operator with new hypergeometric function has been discussed. Similar result obtained by other authors follows as special cases of our findings.

Keywords: Basic analogues of H and I-function, Basic hypergeometric function, Fractional q-integral operators.

1

Introduction:

In the past century, many authors have generalized H-function. In a recent paper, sudland et al.[10] have introduced a generalization of saxena’s I-function[9], which is also a generalization of Fox’s H-function. This function is known as Aleph function. In their paper, sexena and pogany [7] have studied fractional integration formulae for the Aleph functions.

Südland et al. [11] studied the generalized fractional driftless Fokker-Planck equation with power law coefficient. As a result, a special function was found, which is a particular case of the Aleph function. The Aleph function has been defined by means of Mellin- Barnes type contour integrals as

ℵ[𝑧] =

ℵ𝑢𝑚,𝑛_𝑖,𝑣_𝑖,𝜏_𝑖 ;𝑟 𝑧

𝑎𝑗, 𝐴𝑗 _1,𝑛, 𝜏𝑗 𝑎𝑗𝑖, 𝐴𝑗𝑖 _𝑛+1,𝑢

𝑖;𝑟

𝑏𝑗, 𝐵𝑗 _1,𝑚, 𝜏𝑗 𝑏𝑗𝑖, 𝐵𝑗𝑖 _{𝑚 +1,𝑣}

𝑖;𝑟

(1.1)

ℵ[𝑧] = _2𝜋𝜔1 Ω_𝐿 𝑚 ,𝑛𝑢_𝑖,𝑣_𝑖,𝜏_𝑖 ;𝑟(ξ) 𝑧ξ𝑑ξ

For all

z



0

, _{𝜔 = −1 and .}

Ω𝑢𝑚 ,𝑛_𝑖,𝑣_𝑖,𝜏_𝑖 ;𝑟(ξ) = Γ b_j−B_jξ

m

j=1 nj=1Γ 1−aj+Ajξ

𝜏_𝑖 v i_{j=m +1}Γ 1−b_ji+B_jiξ u i_{j=n +1}Γ a_{ji −}A_jiξ

r

i=1

(1.2)

The parameters, 𝑢𝑖, 𝑣𝑖 are non-negative integers satisfying the inequality, and 0 ≤ 𝑛 ≤ 𝑢𝑖 , 1 ≤

𝑚 ≤ 𝑣𝑖 and 𝜏𝑖 ≥ 0 ; i=1, . . . The parameters

Aj, Bj, Aji , Bji are positive real numbers and are complex numbers and a_j , b_j , a_ji , b_jiare complex numbers . The 𝐿 = 𝐿𝜔𝛾 ∞ is a suitable contour of the Mellin-Barnes type in the complex ξ -plane which runs from 𝛾 − 𝜔∞ 𝑡𝑜 𝛾 + 𝜔∞ with ∈ ℜ , such that the poles of Γ bj−Bjξ , j= 1,. . . separating from those of Γ 1 − aj+Ajξ , j=1 , . . . . All the poles of the integrand (1.2) are assumed to be simple, and empty products are interpreted as unity. For the existence conditions,

𝜑𝑙 > 0 , |𝑎𝑟𝑔 𝑧| < (𝜑𝑙 𝜋)/2, for all 𝑖 =

1,2 , … , 𝑟 or,

𝜑𝑙 ≥ 0, |𝑎𝑟𝑔 𝑧| ≤ (𝜑𝑙 𝜋)/2, and

𝑅𝑒( ξ_𝑙 + 1) < 0 for all 𝑙 = 1,2 , … , 𝑟

Where

𝜑𝑙 = 𝐴𝑗 𝑛

𝑗 =1

+ 𝐵𝑗 𝑚

𝑗 =1

− 𝑐𝑙 𝐴𝑗𝑙 𝑢𝑙

𝑗 =𝑛+1

+ 𝐵𝑗𝑙 𝑣𝑙

𝑗 =𝑚+1

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ξ_𝑙 = 𝑛𝑗 =1𝑎𝑗− 𝑗 =1𝑚 𝑏𝑗 + 𝑐𝑙 𝑢𝑗 =𝑛+1𝑙 𝑎𝑗𝑙 −

𝑏𝑗𝑙 𝑣_𝑙

𝑗 =𝑚+1 + 1 2(𝑢𝑙−

𝑣𝑙) 𝑙 = 1,2, ..

The basic analogue of this ℵ-function has been defined by Dutta and Arora [13] in term of Mellin-Barnes type contour integrals in the following manner

ℵ z; q = =

ℵ𝑢𝑚,𝑛_𝑖,𝑣_𝑖,𝜏_𝑖 ;𝑟 𝑧

𝑎𝑗, 𝐴𝑗 _1,𝑛, 𝜏𝑗 𝑎𝑗𝑖, 𝐴𝑗𝑖 _𝑛+1,𝑢

𝑖;𝑟

𝑏𝑗, 𝐵𝑗 _1,𝑚, 𝜏𝑗 𝑏𝑗𝑖, 𝐵𝑗𝑖 _{𝑚 +1,𝑣}

𝑖;𝑟

(1.3)

ℵ[z;q]= 1

2𝜋𝜔 Ω𝑢𝑖,𝑣𝑖,𝜏𝑖 ;𝑟

𝑚 ,𝑛 ₍_ξ_{; q)π 𝑧}ξ

𝐿 𝑑ξ

where 𝜔 = −1

Ω𝑢𝑖,𝑣𝑖,𝜏𝑖 ;𝑟

𝑚 ,𝑛 _ξ_{; q}

= 𝐺(𝑞

𝑏_𝑗− 𝐵_𝑗ξ₎ 𝑚

𝑗 =1 𝑛𝑗 =1𝐺(𝑞1−𝑎𝑗+𝐴𝑗ξ)

{𝜏𝑖 𝑣𝑗 =𝑚+1𝑖 𝐺(𝑞1−𝑏𝑗𝑖+𝐵𝑗𝑖ξ) 𝐺 𝑞𝑎𝑗𝑖−𝐴𝑗𝑖ξ 𝐺 𝑞1−𝑠 𝑠𝑖𝑛𝜋ξ 𝑢𝑖

𝑗 =𝑛+1 }

𝑟 𝑖=1

(1.4)

𝐺 (𝑞𝛿_{) = {}∞ _{(1 − 𝑞}𝛿+𝑗₎

𝑗 =0 }−1 = _(𝑞𝛿1;𝑞)_∞ (1.5)

The parameters,𝑢𝑖, 𝑣𝑖 are non-negative integers

0 ≤ 𝑛 ≤ 𝑢 , 1 ≤

𝑚 ≤ 𝑣𝑖 and 𝜏𝑖 ≥ 0 ; i=1, . . . is finite.The

parameters Aj, Bj, Aji , Bji are positive real

numbers and are complex numbers and

aj , bj , aji , bjiare complex numbers . The 𝐶 =

𝐶𝜔𝛾 ∞ is a suitable contour of the Mellin-Barnes

type in the complex ξ -plane which runs from

𝛾 − 𝜔∞ to 𝛾 + 𝜔∞ with ∈ ℜ , such that the poles

of 𝐺(𝑞𝑏𝑗− 𝐵𝑗ξ₎_{separating from those of}

𝐺(𝑞1−𝑎_𝑗+𝐴_𝑗ξ₎ _{,j=1,. . . . All the poles of the} integrand (3.1.4) are assumed to be simple and empty products are interpreted as unity. The integral converges if ℜ[ξlog z − log sin 𝜋ξ]<0 for large value of |ξ | on the contour, that is if

|arg 𝑧 − 𝜌2𝜌1−1log |𝑧||<𝜋 , where, 0<|q|< 1,

log 𝑞 = −𝜌 = −(𝜌1+ 𝜌𝜌2) , 𝜌 , 𝜌1 are definite

quantities, 𝜌1and 𝜌2 being real.

When all 𝜏_𝑖 = 1; (1.3) yields the q -analogue of the I-function due to Saxena and Kumar [6].

Again, when r = 1, 𝑢𝑖= 𝑢 , 𝑣𝑖 = 𝑣 and 𝑐𝑖 = 1 ; (1.3) yields the q -analogous of the H -function due to Saxena et al. [7].

2 Definitions and Preliminaries:

In this section, we use the following definitions and fundamental facts of basic analogue of special function and integral operator.

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(i) We shall make use of ℵq(z; q) notation for basic analogue of Aleph function using the notation defined earlier by Jain et al. [14] . It is defined as

ℵq(z; q)

= ℵ𝑝𝑚,𝑛_𝑖,𝑞_𝑖,𝜏_𝑖 ;𝑟 𝑧; 𝑞

𝑎𝑗, 𝐴𝑗 _1,𝑛, … . . 𝜏𝑗 𝑎𝑗𝑖, 𝐴𝑗𝑖 _𝑛+1,𝑝

𝑖

𝑏𝑗, 𝐵𝑗 _1,𝑚, … . . 𝜏𝑗 𝑏𝑗𝑖, 𝐵𝑗𝑖 _{𝑚 +1,𝑞}

𝑖

ℵq(z; q) = _2𝜋𝑖1 Ω_𝐿 𝑚 ,𝑛𝑝_𝑖,𝑞_𝑖,𝜏_𝑖 ;𝑟 𝑠; 𝑞 𝑧𝑠 𝑑𝑠 (2.1)

Ω𝑝𝑚 ,𝑛_𝑖,𝑞_𝑖,𝜏_𝑖 ;𝑟 𝑠; 𝑞

= Γq bj−Bjs

m j=1

𝜏𝑖 qj=m+1i Γq 1 − bji+Bjis r

i=1

× Γq 1 − aj+Ajs

Γq aji−Ajis × 𝛤𝑞𝑠 × 𝛤𝑞(1 − 𝑠) 𝑠𝑖𝑛𝜋𝑠 p_i

j=n+1

And

z



0

,i 1.

Here 𝑎𝑖, 𝑏𝑗ϵC > 0; 𝑝𝑖 > 0 , 𝑞𝑗 > 0; 1 +

𝐵𝑗− 𝑝𝑖=1𝐴𝑖 𝑞

𝑗 =1 ≥ 0; 𝑎ϵR for suitably bounded value of |z|.

The integral

converges if Re[slog(z) – log sinπs] < 0, on the contour C, where 0<|q|< 1.

(ii) The fractional q-calculus is the q-extension of the ordinary calculus. Agarwal [1], introduced the q-basic analogue of Kober fractional integral operator in the following form:

I_qη,μf 𝑥 = x−η −µ

Γq(µ) (𝑥 − qt)µ−1t

η_{f t} x

0 dq t

(2.2)

𝑤𝑕𝑒𝑟𝑒 𝑅𝑒 𝜇 > 0, 𝑞 < 1,

(iii) In the theory of q-calculus, 0 < |𝑞| < 1, the q-shifted factorial is defined as

[𝑎; 𝑞]𝑘 =

1 − 𝑎𝑞𝑗 𝑘−1

𝑗 =0 , 𝑖𝑓 𝑘 > 0

(1 − 𝑎𝑞𝑗₎ ∞

𝑗 =0 , 𝑖𝑓 𝑘 ⤏ ∞

(2.3)

Moreover, [𝑎; 𝑞]0 = 1, 𝑎𝑛𝑑 𝑖𝑓 𝑘 = 0 Or equivalently

𝑎; 𝑞 𝑘=_𝑎𝑞 𝑎;𝑞 𝑘_;𝑞∞ ∞

Further if α is any complex number, then the

definition can be stated as follows

(𝑎; 𝑞)𝛼 = _(𝑎𝑞(𝑎;𝑞)𝛼;𝑞)∞_∞

(𝑎; q)∞ = Γ_q(α)

(1−q)αG(q)α (2.4)

The q-analogue of power function is defined and denoted as

(𝑎 − b)α = 𝑎α 𝑏_𝑎; 𝑞 = 𝑎α

1−b_𝑎 qj 1−b_𝑎 qj+α

∞

j=0 =

𝑎α

𝑏 𝑎; 𝑞 ∞

(qα b 𝑎; q)∞

(2.5)

The q- gamma function is defined by

𝛤𝑞 𝛼 =𝐺(𝑞

𝛼₎

𝐺(𝑞) (1 − 𝑞)

1−𝛼_{= (1 − 𝑞)} 𝛼−1(1 −

𝑞)1−𝛼_{; 𝛼 ∈ R(0, −1, −2, … )}_(2.6)

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The q-derivatives of a function f(x) is given as follows:

Dqf 𝑥 = f 𝑥 −f(q𝑥)_(1−q)𝑥 , 𝑥 ≠0) and (Dqf) 0=0

(2.7)

Dq ⤏_{d x}d as q ⤏1.

We have

Dqn(xµ) = Γ_q(µ+1) Γ_q(µ−n+1)x

µ−n _{, R(µ) + 1 > 0}

The q-integral of a function is defined as :

𝑓 𝑡 ₀𝑥 𝑑𝑞 𝑡 = 𝑥(1 − 𝑞) ∞𝑘=0𝑞𝑘𝑓(𝑥𝑞𝑘)

(2.8)

𝑓 𝑡 _𝑥∞ 𝑑𝑞 𝑡 = 𝑥(1 − 𝑞) ∞𝑘=1𝑞−𝑘𝑓(𝑥𝑞−𝑘)

(2.9)

𝑓 𝑡 ₀∞ 𝑑𝑞 𝑡 = (1 − 𝑞) ∞𝑘=−∞𝑞𝑘𝑓(𝑥𝑞𝑘) (2.10)

The q- binomial theorem is given as follows

1 ɸ 0 𝛼; 𝑞; 𝑥 = (𝑎;𝑞)𝑛

(𝑞;𝑞)𝑛𝑧

𝑛 ₌ ∞

𝑛=0 (𝛼𝑥 ;𝑞)_(𝑥;𝑞)∞

∞

(2.11)

The q- analogues of Gauss summation theorem is given by

2 ɸ 1 𝑞𝑎_{; 𝑞}𝑏_{; 𝑞}𝑐_{; 𝑞; 𝑞}𝑐−𝑎−𝑏₌𝛤𝑞 𝑐 𝛤𝑞 𝑐−𝑎−𝑏

𝛤_𝑞(𝑐−𝑎)𝛤_𝑞(𝑐−𝑏) (2.12)

Agarwal [1] introduced the q-analogue of the Reimann-Liouville fractional integral operator as follows.

Iµqf(xµ)= _Γ_q1_(µ) (1 − qy)₀∞ µ−1f y dq y

where µ is an arbitrary order of integration such that Re (μ)> 0. From equation (2.8) and (2.13),

we get

𝐼𝑞µ𝑓(𝑥)= 𝑥µ(1 − 𝑞)𝜇 𝑞

𝑘_𝑞𝜇_{; 𝑞}_𝑘

𝑞;𝑞 _𝑘 ∞

𝑘=0 𝑓 𝑥𝑞𝑘

(2.14)

Agarwal [1], introduced the basic analogue of Kober fractional operator as follows.

Iqη,μf 𝑥 =x

−η −µ

Γ_q(µ) (𝑥 − qt)µ−1t η_{f t} x

0 dq t

(2.15)

Again from equation (2.12) and (2.14), we get

Iqη,μf(𝑥) = (1 − q)μ

qk(1+η )_(qμ_{; q)}_k

(q;q)_k ∞

k=0 f(𝑥qk)

(2.16)

For the basic concept of q-calculus we refer to reader to [3]

3 Main Result:

In this section, we will establish the following fractional q-integral operator of Kober type involving basic analogue of Aleph-function.

Theorem (3.1) Let ℜ 𝛼 > 0, q < 1 𝑎𝑛𝑑 I_qη,µ be the q-analogue of the Kober fractional integral operator (2.2) ,then following results holds:

Iη,µq zρ−1ℵ𝑚,𝑛𝑝_𝑖,𝑞_𝑖,𝜏_𝑖 ;𝑟 ςzλ; q

𝑎𝑗, 𝐴𝑗 _1,𝑛; [𝜏𝑗 𝑎𝑗𝑖, 𝐴𝑗𝑖 ]𝑛+1,𝑝_𝑖

𝑏𝑗, 𝐵𝑗 _1,𝑚; [𝜏𝑗 𝑏𝑗𝑖, 𝐵𝑗𝑖 ]𝑚 +1,𝑞_𝑖

= (1 − q)μ_{× z}ρ−1 ℵ𝑝𝑚,𝑛_𝑖+1,𝑞_𝑖+1,𝜏_𝑖 ;𝑟 ςzλ;

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q

1 − η − ρ, λ aj, Aj _1,n;

bj, Bj _1,m;

[𝜏𝑗 𝑎𝑗𝑖, 𝐴𝑗𝑖 ]𝑛+1,𝑝_𝑖

[𝜏𝑗 𝑏𝑗𝑖, 𝐵𝑗𝑖 ]𝑚 +1,𝑞_𝑖 1 − μ − η − ρ, λ

; for λ ≥ 0

= (1 − q)μ_{× z}ρ−1_{ℵ

𝑝_𝑖+1,𝑞_𝑖+1,𝜏_𝑖 ;𝑟 𝑚,𝑛

× ςzλ_{; q} aj, Aj 1,n;

η + ρ, −λ bj, Bj _1,m;

[𝜏𝑗 𝑎𝑗𝑖, 𝐴𝑗𝑖 ]𝑛+1,𝑝𝑖 μ + η + ρ, −λ [𝜏𝑗 𝑏𝑗𝑖, 𝐵𝑗𝑖 ]𝑚 +1,𝑞𝑖

; for λ < 0

Where ℜ 𝜉 log η − log sin 𝜋𝜉 < 0 and ρ > 0 .

Proof: To prove theorem (3.1) when λ ≥ 0, we

apply (2.16) and (2.1) to the left side, we get

Iqη,µ zρ−1ℵ𝑚,𝑛𝑝_𝑖,𝑞_𝑖,𝜏_𝑖 ;𝑟 ςzλ; q

𝑏𝑗, 𝐵𝑗 _1,𝑚; [𝜏𝑗 𝑏𝑗𝑖, 𝐵𝑗𝑖 ]𝑚+1,𝑞_𝑖

= (1 − q)μ_× 1

2πi

qk 1+η _qμ_{; q} k

q; q k ∞

k=0

(zqk₎ρ−1 L

× Γq bj−Bjs m

j=1 j=1n Γq 1 − aj+Ajs (ςzλqkλ)𝑠𝑑𝑠

𝜏𝑖 qij=m+1Γq 1 − bji+ Bjis pij=n+1Γq aji−Ajis × 𝛤𝑞𝑠 × 𝛤𝑞(1 − 𝑠) 𝑠𝑖𝑛𝜋𝑠 r

i=1

This implies

=(1 − q)μ× zρ−1

1 2πi

Γq bj−Bjs m

j=1 nj=1Γq 1 − aj+Ajs (ςzλ)𝑠

𝜏𝑖 Γq 1 − bji+ Bjis pij=n+1Γq aji−Ajis × 𝛤𝑞𝑠 × 𝛤𝑞(1 − 𝑠) 𝑠𝑖𝑛𝜋𝑠 qi

j=m+1 r

i=1 L

× ɸ1 0 [qµ; q ; q(ρ+η+λs)]ds

On summing the inner₁ɸ₀ series with help of (2.10), we get

(1 − q)μ

× zρ−1 1

2πi

Γq bj− βjs m

j=1 × nj=1Γq 1 − aj+ αjs

qij=m+1Γq1 − bji+ βjis) r

i=1 L

× Γq ρ + η + λs (ςzλ)𝑠

Γ_q(a_ji− α_jis) × 𝛤_𝑞𝑠 × 𝛤_𝑞(1 − 𝑠) 𝑠𝑖𝑛𝜋𝑠Γ_q ρ + μ + η + λs pi

j=n+1

ds

Now using (2.1) we get

Iqη,µ zρ−1ℵ𝑝𝑖,𝑞𝑖,𝜏𝑖 ;𝑟

𝑚,𝑛 _ςzλ_{; q} 𝑎𝑗, 𝐴𝑗 1,𝑛; [𝜏𝑗 𝑎𝑗𝑖, 𝐴𝑗𝑖 ]𝑛+1,𝑝𝑖

=

(1 − q)

μ

× z

ρ−1

ℵ

_𝑝 𝑚 ,𝑛 𝑖+1,𝑞𝑖+1;𝜏𝑖,𝑟

× ςz

λ_{; q} 1 − η − ρ, λ aj, Aj 1,n; bj, Bj _1,m;

[𝜏𝑗 𝑎𝑗𝑖, 𝐴𝑗𝑖 ]𝑛+1,𝑝𝑖

[𝜏𝑗 𝑏𝑗𝑖, 𝐵𝑗𝑖 ]𝑚+1,𝑞𝑖 1 − μ − η − ρ, λ

, for λ ≥ 0

Also when λ < 0 in theorem (3.1) we get a known result due to Yadav etal [15] as follows

Iqη,µ zρ−1ℵ𝑝𝑖,𝑞𝑖,𝜏𝑖 ;𝑟

𝑚,𝑛 _ςzλ_{; q} 𝑎𝑗, 𝐴𝑗 1,𝑛; [𝜏𝑗 𝑎𝑗𝑖, 𝐴𝑗𝑖 ]𝑛+1,𝑝𝑖

= (1 − q)

μ

_{× z}

ρ−1

_ℵ

𝑚, 𝑛

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ςzλ_{; q} aj, Aj 1,n; η + ρ, −λ bj, Bj _1,m;

[𝜏𝑗 𝑎𝑗𝑖, 𝐴𝑗𝑖 ]𝑛+1,𝑝𝑖 μ + η + ρ, −λ [𝜏𝑗 𝑏𝑗𝑖, 𝐵𝑗𝑖 ]𝑚+1,𝑞𝑖

, for λ < 0

4 Special Cases:

The q-extension of Aleph function defined by (1.3) in terms of Mellin-Barnes type of basic integrals is most general in nature, which includes a number of basic analogues of special functions. In this section we discuss only the case involving 𝐼𝑞(. ) function

and 𝐻_𝑞(. ) function.

(i) if we set 𝜏_𝑗 = 1, i = 1, . .. in above theorem

we obtain the following formula for 𝐼𝑞(. ) function

Corollary 4.1 Let ℜ 𝛼 > 0, |q| < 1, then the following results holds:

Iqη,µ zρ−1𝐼𝑝𝑚 ,𝑛_𝑖,𝑞_𝑖 ;𝑟 ςzλ; q 𝑎_𝑗,𝐴_𝑗

1,𝑛; 𝑎𝑗𝑖,𝐴𝑗𝑖 _{𝑛 +1,𝑝 𝑖}

𝑏_𝑗,𝐵_𝑗 _1,𝑚; 𝑏_𝑗𝑖,𝐵_𝑗𝑖

𝑚 +1,𝑞𝑖

= (1 − q)μ_{× z}ρ−1_𝐼

𝑝_𝑖+1,𝑞_𝑖+1 ;𝑟 𝑚,𝑛

ςzλ_{; q} 1 − η − ρ, λ aj, Aj 1,n; bj, Bj _1,m;

𝑎𝑗𝑖, 𝐴𝑗𝑖 _{𝑛+1,𝑝𝑖} 𝑏𝑗𝑖, 𝐵𝑗𝑖 _{𝑚+1,𝑞𝑖} 1 − μ − η − ρ, λ

; for λ ≥ 0

= (1 − q)μ_{× z}ρ−1_𝐼

𝑝_𝑖+1,𝑞_𝑖+1 ;𝑟 𝑚,𝑛

ςzλ_{; q} 1 − η − ρ, λ aj, Aj 1,n; bj, Bj _1,m;

𝑎𝑗𝑖, 𝐴𝑗𝑖 _{𝑛+1,𝑝𝑖} 𝑏𝑗𝑖, 𝐵𝑗𝑖 _{𝑚+1,𝑞𝑖} 1 − μ − η − ρ, λ

(ii) if we set r = 1, 𝜏𝑗 = 1, , 𝑝𝑖 = p, q𝑖 = q 𝑎𝑗𝑖 =

aj 𝑏𝑗𝑖 = bj , 𝐴𝑗𝑖 = Aj , 𝐵𝑗𝑖 = Bj in above theorem

we obtain the following formula for 𝐻𝑞(. )

function.

Corollary 4.2 Let ℜ 𝛼 > 0, |q| < 1, then the following results holds:

Iqη,µ zρ−1𝐻𝑝,𝑞𝑚 ,𝑛 ςzλ; q

𝑎𝑗, 𝐴𝑗 _1,𝑝

𝑏𝑗, 𝐵𝑗 _1,𝑞

= (1 − q)μ

× zρ−1

𝐻𝑝+1,𝑞+1𝑚 ,𝑛 ςzλ; q

1 − η − ρ, λ aj, Aj _1,p

bj, Bj _1,q, 1 − μ − η − ρ, λ

; for λ ≥ 0

= (1 − q)μ

× zρ−1

𝐻_𝑝𝑚 ,𝑛_𝑖_+1,𝑞_𝑖_+1,;𝑟_ςzλ_{; q} aj, Aj 1,p μ + η + ρ, −λ

η + ρ, −λ bj, Bj _1,q

; for λ < 0

5 Conclusion:

(7)

[12]. Thus these results can be applied to various problems of mathematical physics.

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