On various formulas with q-integralsand their applications to q-hypergeometric functions

Vol. 13, No. 5, 2020, 1241-1259 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global

Special Issue Dedicated to Professor Hari M. Srivastava On the Occasion of his 80th Birthday On various formulas with q-integralsand their

applications to q-hypergeometric functions

Thomas Ernst

Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 Uppsala, Sweden

Abstract. We present three q-Taylor formulas with q-integral remainder. The two last proofs require a slight rearrangement by a well-known formula. The first formula has been given in different form by Annaby and Mansour. We give concise proofs for q-analogues of Eulerian integral formulas for general q-hypergeometric functions corresponding to Erd´elyi, and for two of Srivastavas triple hypergeometric functions and other functions. All proofs are made in a similar style by using q-integration. We find some new formulas for fractional q-integrals including a series expansion.

In the same way, the operator formulas by Srivastava and Manocha find a natural generalization.

2020 Mathematics Subject Classifications: 33D70, 41A58, 33C65, 33D15

Key Words and Phrases: q-Taylor formulas with q-integral remainder, q-hypergeometric func- tion, q-Eulerian integral, fractional q-integral

1. Introduction 1.1. General

The aim of this paper is to continue the investigation of single and multiple q-hypergeometric series in the spirit of our book [9] and our paper [12] on the q-Lauricella functions. The fractional q-integrals and direct computations of q-integrals lead to similar results. We refer to previous papers with respect to convergence regions. By quoting Erd´elyi and Feld- heim, we have managed to save these hypergeometric formulas from oblivion; our proofs of their formulas are quite similar, although these authors never wrote down their proofs. In the same style, Charles Cailler in 1920 [3], [20, p. 242] and Kamp´e de F´eriet in 1922 [19, p.

DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3755 Email address: thomas@math.uu.se (T. Ernst)

https://www.ejpam.com 1241 2020c EJPAM All rights reserved.

26] published a hypergeometric formula and a fractional integral formula, respectively. We will q-deform the latter formula, and show that it is a special case of an operator formula by Srivastava and Manocha [23, p.289 (18)]. Instead, Koschmieder [20, p. 252], for the first time, published a formula with a double Eulerian integral for a fractional derivative of a double hypergeometric series times power functions. By a standard procedure we are able to prove a q-analogue of Koschmieders formula. In the end, we use a generalization of the q-binomial theorem to find a q-analogue of an operator formula by Srivastava and Manocha [23, p.306].

This paper is organized as follows: In section 1.1 we prove the three q-Taylor formulas by using q-integration by parts. Erd´elyi formulas and fractional q-integrals are dicussed in section 1.1. Finally, in section ?? we consider similar, more complicated formulas in the spirit of Srivastava and Manocha [23]. \section{Threeq-Taylor formulas In a recent paper [2] Annaby and Mansour have found two q-Taylor formulas with q-integral remainder: [9, (8.90)] and a formula similar to (2). We are going to prove (2) and two other q-Taylor formulas which use the two types of q-addition. All proofs use q-integration by parts. To be able to work freely, we consider functions in C[[x]]. We use the following definition.

Definition 1.

Pn,q(x, a) ≡

n−1

Y

m=0

(x + aq^m), n = 1, 2, . . . . (1) A different form of the following theorem, without remainder, occurred in Al-Salam, Verma [1, 2.2] [2, p. 480 4.6].

Theorem 1. Let 0 < |q| < 1 and let f be n times q-differentiable in the closed interval [a, x]. Then the following generalization of Jackson’s formula holds for n = 1, 2, . . .:

f (x) =

n−1

X

k=0

(−1)^kq⁻(^k2)P_k,q(a, −x)

{k}_q! (D^k_qf )(aq^−k)+

Z x t=a

(−1)ⁿ⁻¹q⁻(ⁿ2)P_n−1,q(t, −x)

{n − 1}_q! (Dⁿ_qf )(tq⁻ⁿ) dq(t).

(2)

Proof. We use q-integration by parts We start with f (x) = f (a) +

Z x t=a

Dqf (t)Dq,t(t − x) dq(t), (3) which follows from the definition of q-integral. At the next step we obtain

f (x) = f (a) +D_qf (tq⁻¹)(t − x)t=x t=a−

Z x t=a

q⁻¹(t − x)D²_qf (tq⁻¹) dq(t). (4) We proceed with

f (x) = f (a) + Dqf (aq⁻¹)(x − a) −



D²_qf (tq⁻²) q⁻¹

{2}_q!(t²− xt(1 + q) + qx²)

t=x t=a

+ Z _x

t=a

D³_qf (tq⁻²)q⁻³P2,q(t, −x) {2}_q! d_q(t).

(5)

In the third step we obtain

f (x) = f (a) − Dqf (aq⁻¹)P1,q(a, −x) + D²_qf (aq⁻²) 1

{2}_q!P2,q(a, −x)+



D³_qf (tq⁻³)q⁻³P_3,q(t, −x) {3}_q!

t=x t=a

− Z x

t=a

D⁴_qf (tq⁻⁴)q⁻⁶P_3,q(t, −x) {3}_q! d_q(t).

(6)

We can continue this process forever.

We are going to use the following formula in the last proofs:

Theorem 2. von Gr¨uson [22, S. 36] 1814, [9, 2.19].

m

X

n=0

(−1)ⁿm n



q

q(ⁿ²)uⁿ= (u; q)m. (7)

Our next aim is to find q-Taylor expansions with q-integral remainder for formulas corresponding to Nalli–Ward and Jackson respectively. These formulas are (18) and (22).

We first prove preliminary lemmata (8) and (13), and then show by (7) that these are equivalent to the formulas we want to prove.

Lemma 1.

F (x ⊕_qy) = F (x) +

n−1

X

k=1

y^k

{k}_q!(−1)^k+1q(^k2)D_q,y^k F (x ⊕_qy)+

Z y t=0

D_q,tⁿ [F (x ⊕qt)] (−t)ⁿ⁻¹

{n − 1}_q!q(ⁿ²) d_q(t).

(8)

Proof. The proof is very straightforward, at each step we only use integration by parts.

We start with

(x ⊕qy)^m = x^m+ Z y

t=0

Dq,t[(x ⊕qt)^m] Dq,t(t) dq(t), (9) which follows from the definition of q-integral. At the next step we obtain

(x ⊕qy)^m= x^m+ [Dq,t[(x ⊕qt)^m] t]^t=y_t=0− Z y

t=0

qtD_q,t² [(x ⊕qt)^m] dq(t). (10) We proceed with

(x ⊕_qy)^m = x^m+ yD_q,y((x ⊕_qy)^m) −



D²_q,t[(x ⊕_qt)^m] qt² {2}_q!

t=y t=0

+ Z y

t=0

D_q,t³ [(x ⊕qt)^m] q³t² {2}_q!dq(t).

(11)

In the third step we obtain

(x ⊕_qy)^m= x^m+ yD_q,y(x ⊕_qy)^m− D_q,y² [(x ⊕_qy)^m] qy² {2}_q!+



D_q,t³ [(x ⊕qt)^m] q³t³ {3}_q!

t=y t=0

− Z y

t=0

D_q,t⁴ [(x ⊕qt)^m] q⁶t³ {3}_q!dq(t).

(12)

We can continue this process forever.

Lemma 2.

F (x
qy) = F (x) +

n−1

X

k=1

y^k

{k}_q!(−1)^k+1q(^k2)D_q,y^k F (x
qy)+

Z y t=0

D_q,tⁿ [F (x
qt)] (−t)ⁿ⁻¹

{n − 1}_q!q(ⁿ²) d_q(t).

(13)

Proof. The proof is almost the same as the previous one. We start with (x
qy)^m = x^m+

Z y t=0

D_q,t[(x
qt)^m] D_q,t(t) d_q(t), (14) which follows from the definition of q-integral. At the next step we obtain

(x
qy)^m= x^m+ [Dq,t[(x
qt)^m] t]^t=y_t=0− Z y

t=0

qtD_q,t² [(x
qt)^m] dq(t). (15) We proceed with

(x
qy)^m = x^m+ yD_q,y((x
qy)^m) −



D²_q,t[(x
qt)^m] qt² {2}_q!

t=y t=0

+ Z y

t=0

D_q,t³ [(x
qt)^m] q³t² {2}_q!dq(t).

(16)

In the third step we obtain

(x
qy)^m= x^m+ yD_q,y(x
qy)^m− D_q,y² [(x
qy)^m] qy² {2}_q!+



D_q,t³ [(x
qt)^m] q³t³ {3}_q!

t=y t=0

− Z y

t=0

D_q,t⁴ [(x
qt)^m] q⁶t³ {3}_q!dq(t).

(17)

We can continue this process forever.

Theorem 3. Compare with the Nalli–Ward q-Taylor formula [9], which is obtained by letting n → ∞.

F (x ⊕qy) =

n−1

X

k=0

y^k

{k}_q!D^k_qF (x)+

Z _y

t=0

Dⁿ_q,t[F (x ⊕_qt)] (−t)ⁿ⁻¹

{n − 1}_q!q(ⁿ²) d_q(t).

(18)

Proof. We show that this is equivalent with (8). By putting F (x) = x^mit would suffice to prove that

m

X

n=1

yⁿ

{n}_q!D_qⁿx^m=

m

X

k=1

y^k

{k}_q!(−1)^k+1q(^k2)D^k_q,y(x ⊕_qy)^m. (19) This is equivalent to the formula

m

X

n=1

yⁿ

{n}_q!{m − n + 1}_n,qx^m−n=

m

X

k=1

y^k

{k}_q!(−1)^k+1q(^k2){m − k + 1}_k,q

m−k

X

l=0

{m − k}_q!

{m − k − l}_q!{l}q!x^ly^m−k−l.

(20)

By equating the corresponding exponents for x and y, and thus putting n = m − l we obtain

{l + 1}_m−l,q {n}_q! =

m−l

X

k=1

(−1)^k+1q(^k2){m − k + 1}_k,q {k}_q!

{m − k}_q!

{m − k − l}_q!{l}_q!. (21) After simplification we see that this is equivalent to (7) for the special case u = 1.

Theorem 4. Compare with the second Jackson q-Taylor formula, which is obtained by letting n → ∞.

F (x
qy) =

n−1

X

k=0

y^k

{k}_q!q(^k2)D_q^kF (x)+

Z y t=0

Dⁿ_q,t[F (x
qt)] (−t)ⁿ⁻¹

{n − 1}_q!q(ⁿ²) d_q(t).

(22)

Proof. We show that this is equivalent with (13). By putting F (x) = x^m it would suffice to prove that

m

X

n=1

yⁿ

{n}_q!q(ⁿ²)D_qⁿx^m=

m

X

k=1

y^k

{k}_q!(−1)^k+1q(^k2)D^k_q,y(x
qy)^m. (23) This is equivalent to the formula

m

X

n=1

yⁿ

{n}_q!{m − n + 1}_n,qq(ⁿ²)x^m−n=

m

X

k=1

y^k

{k}_q!(−1)^k+1{m − k + 1}_k,q

m−k

X

l=0

{m − k}_q!

{m − k − l}_q!{l}_q!x^ly^m−k−l QE



k²− k + k(m − k − l) + (m − k − l)²− (m − k − l) 2

 .

(24)

By equating the corresponding exponents for x and y, and thus putting n = m − l we obtain

{l + 1}_m−l,q

{n}_q! qm2+l2−2ml+l−m

2 =

m−l

X

k=1

(−1)^k+1{m − k + 1}_k,q {k}_q!

{m − k}_q! {m − k − l}_q!{l}q! QE



k²− k + k(m − k − l) + (m²+ k²+ l²− 2mk − 2ml + 2kl) − (m − k − l) 2

 .

(25)

After simplification we see that this is equivalent to (7) for the special case u = 1.

\section{Erd\’elyiformulas and fractional q-integrals In this section we have col- lected several q-Euler integral expressions for the function ₂φ₁(α, β; γ|q; z) and related formulas. Each of these q-Euler integral formulas have a prefactor Γq function. The re- strictions for the parameters in these prefactors are the same as in the original formula, i.e. Re(parameters) > 0. For the notation, see our book [9].

Theorem 5. A q-analogue of Erd´elyi [7, (2.6) p. 270]. Assume that ~α , ~µ and ~s are vectors of length m and ~β and ~γ are vectors of length p + 1 − m and p, respectively, where p + 1 > m. Then we have the q-Euler integral representation

p+1φp

 ~α, ~β

~ γ

q; x



= Γ_q

 ~µ

~ α,µ − α~

 Z ^~1

~s=~0

~

s^α−1~ (q~s; q)_{µ−α−1 p+1~} φ_p

 ~µ, ~β

~γ    

q; x~s

 d_q(~s).

(26)

Proof. We compute the right hand side:

RHS^by[9,6.54]

= Γq

 ~µ

~ α,µ − α~

 ∞

X

n=0

~

∞

X

~k=~0

h~µ, ~β; qinxⁿ

h1, ~γ; qi_n (1 − q)^mq^k(α+n)h ~1 + k; qi_µ−α−1~

by[9,6.8,6.10]

= Γq

 ~µ

~ α,µ − α~

 ∞

X

n=0

h~µ, ~β; qinxⁿ

h1, ~γ; qi_n (1 − q)^m

~

∞

X

~k=~0

q^k(α+n)h ~µ − α; qi_~kh~1; qi_∞~ h~1; qi_~kh ~µ − α; qi∞_~ by[9,7.27]

= Γq

 ~µ

~ α,µ − α~

 ^∞ X

n=0

h~µ, ~β; qinxⁿ

h1, ~γ; qi_n (1 − q)^m h ~µ + n, 1; qi∞_~

h ~α + n,µ − α; qi~ _∞~

by[9,1.45,1.46]

= LHS.

(27)

Theorem 6. A q-analogue of Erd´elyi [7, (5.2) p. 273]. Assume that ~γ , ~δ and ~s are vectors of length m and ~α and ~β are vectors of length p + 1 and p − m, respectively where p > m. Then we have the q-Euler integral representation

p+1φ_p

 α~

~γ, ~β    

q; x



= Γq

 ~γ

~δ,γ − δ~

 Z ^~1

~ s=~0

~s^δ−1~ (q~s; q)_{γ−δ−1 p+1~} φp

 α~

~δ, ~β    

q; x~s

 dq(~s).

(28)

Proof. We compute the right hand side:

RHS^by[9,6.54]

= Γ_q

 ~γ

~δ,γ − δ~

 ^∞ X

n=0

∞~

X

~k=~0

h~α; qinxⁿ

h1, ~δ, ~β; qi_n(1 − q)^mq^k(δ+n)h ~1 + k; qi_γ−δ−1~

by[9,6.8,6.10]

= Γq

 ~γ

~δ,γ − δ~

 ∞

X

n=0

~

∞

X

~k=~0

h~α; qinxⁿ h1, ~δ, ~β; qin

(1 − q)^mq^k(δ+n)h ~γ − δ; qi_~kh~1; qi_∞~ h~1; qi_~kh ~γ − δ; qi∞_~ by[9,7.27]

= Γ_q

 ~γ

~δ,γ − δ~

 ^∞ X

n=0

h~α; qi_nxⁿ

h1, ~δ, ~β; qi_n(1 − q)^m h ~γ + n, 1; qi∞_~

h ~δ + n,γ − δ; qi~ _∞~

by[9,1.45,1.46]

= LHS.

(29)

We can now combine the two previous theorems.

Theorem 7. A q-analogue of Erd´elyi [7, (6.1) p. 274]. Assume that ~µ , ~α and ~s are vectors of length m and ~γ, ~ and ~t are vectors of length n, where m + n = p + 1.

Then we have the q-Euler integral representation

p+1φp

"

~ α, ~β

~ γ, ~δ

q; x

#

= Γq

 ~µ, ~γ

~

α,µ − α, ~,~ γ − ~

 Z ~1

~s=~0

~

s^α−1~ (q~s; q)_µ−α−1~

× Z ~1

~t=~0

~t^−1~ (q~t; q)_{γ−−1 p+1~} φp

"

~ µ, ~β

~, ~δ     

q; x~s~t

#

dq(~s) dq(~t).

(30)

Proof. Put

D ≡ Γq

 ~µ, ~γ

~

α,µ − α, ~,~ γ − ~

 ^∞ X

i=0

h~µ, ~β; qiixⁱ

h1, ~, ~δ; qi_i(1 − q)^m+n. (31) Then we have

RHS^by[9,6.54]

= D

∞~

X

~k=~0

q^k(α+i)h ~1 + k; qi_µ−α−1~

∞~

X

~l=~0

q^l(+i)h ~1 + l; qi_γ−−1~

by[9,6.8,6.10]

= D

∞~

X

~k=~0

q^k(α+i)h ~µ − α; qi_~kh~1; qi_∞~ h~1; qi_~kh ~µ − α; qi_∞~

∞~

X

~l=~0

q^l(+i)h ~γ − ; qi_~lh~1; qi_∞~ h~1; qi_~lh ~γ − ; qi_∞~

by[9,7.27]

= D h ~γ + i,µ + i, 1, 1; qi~ ∞_~

h ~γ − , ~ + i,α + i,~ µ − α; qi~ _∞~

by[9,1.45,1.46]

= LHS.

(32)

We quote a few theorems from our book [9].

Lemma 3. Linear substitution in a q-integral [9, 6.64].

Z x 0

f (t, q) dq(t) = a Z ^x

a

0

f (at, q) dq(t). (33)

Definition 2. [9, 8.116]

Pα,q(x, a) ≡ x^α (^a_x; q)∞

(_x^aq^α; q)∞

, ^a_x 6= q^−m−α, m = 0, 1, . . . . (34) We remark that a totally different approach to the following formulas was made in [2, p.124-125].

Definition 3. [9, 8.117] The fractional q-integral is defined in the following way, ν ∈ C:

D^−ν_q f (x) ≡ 1 Γ_q(ν)

Z x 0

Pν−1,q(x, qt)f (t) dq(t). (35) We infer that

Theorem 8. [9, 8.118]

D^−ν_q x^µ= Γq(µ + 1)

Γ_q(µ + ν + 1)x^µ+ν. (36)

Theorem 9. A q-analogue of Mathai, Haubold [21, 3.2.1], Holmgren [16, p. 3 (4)], Kamp´e de F´eriet [19, p. 200]. Assume that n − 1 < Re(−ν) < n. Then the fractional q-integral (35) can also be expressed as

D^−ν_q f (x) = 1

Γq(n + ν)Dⁿ_q,x Z x

a

P_n+ν−1,q(x, qt)f (t) d_q(t). (37)

Proof. The proof is by induction.

1

Γ_q(1 + ν)D_q,x Z _x

a

P_ν,q(x, qt)f (t) d_q(t)

= 1 − q

Γq(1 + ν)x(q − 1)

∞

X

n=0

qⁿ



(qx)^ν+1 (qⁿ⁺¹; q)∞

(q^n+1+ν; q)∞

f (xqⁿ⁺¹)

−a(qx)^ν (^a_xqⁿ; q)∞

(_x^aq^n+ν; q)∞

f (aqⁿ)

−x^ν+1 (qⁿ⁺¹; q)∞

(q^n+1+ν; q)∞

f (xqⁿ) + ax^ν (^a_xqⁿ⁺¹; q)∞

(^a_xq^n+1+ν; q)∞

f (aqⁿ)



= x^ν(1 − q) Γq(ν)

"_∞ X

n=0

qⁿ(qⁿ⁺¹; q)∞

(q^n+ν; q)∞

f (xqⁿ)1 − q^n+ν 1 − q^ν

−

∞

X

n=1

q^n+ν (qⁿ; q)∞

(q^n+ν; q)∞

f (xqⁿ)1 − qⁿ 1 − q^ν

#

+ ax^ν−1(1 − q) Γq(ν)(1 − q^ν)

∞

X

n=0

qⁿ(^a_xqⁿ⁺¹; q)∞

(^a_xq^n+ν; q)∞

f (aqⁿ) h

−(1 − a

xq^n+ν) + q^ν(1 − a xqⁿ)

i

= x^ν−1(1 − q) Γ_q(ν)

∞

X

n=0

qⁿ x(qⁿ; q)∞

(q^n+ν; q)∞

f (xqⁿ) − a(^a_xqⁿ⁺¹; q)∞

(_x^aq^n+ν; q)∞

f (aqⁿ)



= 1

Γq(ν) Z x

a

Pν−1,q(x, qt)f (t) dq(t).

(38)

We can continue this process to an arbitrary integer n like in the previous q-Taylor ex- pansions.

The following lemma enables a series expansion for D^−ν_q f (x).

Lemma 4.

Dq,t



−Pν,q(x, t) {ν}_q



= Pν−1,q(x, qt). (39)

Theorem 10. A q-analogue of [16, p.8 (19)].

D^−ν_q f (x)

=

k−1

X

j=0

(D^j_qf )(a)Pj+ν,q(x, a)

Γq(ν + j + 1) + 1 Γq(k + ν)

Z x a

P_k+ν−1,q(x, qt)(D^k_qf )(t) d_q(t). (40) Proof. At each step we only use q-integration by parts [9, 6.58] and formula (39). Put D_qv(t) = P_ν−1,q(x, qt), u(t) = f (t). (41) Then

LHS = 1 Γq(ν)



− f (t) {ν}_qx^ν(t

x; q)_ν

x a

+ Z x

a

P_ν,q(x, qt)(D_qf )(t) {ν}_q d_q(t)



. (42)

We can continue this process k times.

Theorem 11. Assume that the convergence region for Φ₂(α; β₁, β₂; , γ₁, γ₂|q; x, y) is [10].

|x| ⊕_q|y| < 1. (43)

A q-analogue of Koschmieder [20, p. 252]. Put F (x, y) ≡ Φ^1:3_1:2

 α : β₁, µ₁, ∞; β₂, µ₂, ∞

∞ : ν₁, τ1; ν2, τ2

q; x, y



. (44)

Then we have the q-Euler integral representation Φ2(α; β1, β2; , γ1, γ2|q; x, y) = Γ_q

 γ₁, γ₂, µ₁, µ₂

ν1, γ1− ν₁, ν2, γ2− ν₂, τ1, τ2



× Z 1

s=0

Z 1 t=0

s^ν1−µ1t^ν2−µ2(qs; q)_γ1−ν1−1(qt; q)_γ2−ν2−1

D^τ_q,s^1−µ1D^τ_q,t^2−µ2s^τ1−1t^τ2−1F (sx, ty) d_q(s) d_q(t).

(45)

Proof. Put

D ≡ Γq

 γ1, γ2, µ1, µ2

ν₁, γ₁− ν₁, ν₂, γ₂− ν₂, τ₁, τ₂



∞

X

m,n=0

hα; qi_m+nhβ₁, µ₁; qi_mhβ₂, µ₂; qi_n h1, ν₁, τ1; qimh1, ν₂, τ2; qin

x^myⁿ.

(46)

We compute the right hand side:

RHS^by[9,8.118]

= D

Z 1 s=0

Z 1 t=0

(qs; q)γ1−ν₁−1(qt; q)γ2−ν₂−1

Γq

 m + τ1, n + τ2, m + µ₁, n + µ₂



s^m+ν1−1t^n+ν2−1dq(s) dq(t)

by[9,6.54]

= D(1 − q)²Γq

 m + τ₁, n + τ₂, m + µ1, n + µ2



∞

X

k,l=0

q^{k(ν1+m)+l(ν2+n)}h1 + k; qi_{γ1−ν1−1}h1 + l; qi_{γ2−ν2−1}

by[9,6.8,6.10]

= D(1 − q)²Γ_q

 m + τ₁, n + τ₂, m + µ₁, n + µ₂



∞

X

k,l=0

q^{k(ν1+m)+l(ν2+n)}hγ₁− ν₁; qikhγ₂− ν₂; qilh1, 1; qi_∞ h1; qi_kh1; qi_lhγ₁− ν₁, γ₂− ν₂; qi∞ by[9,7.27]

= D(1 − q)²Γ_q

 m + τ1, n + τ2, m + µ₁, n + µ₂

 hγ₁+ m, γ2+ n, 1, 1; qi∞

hγ₁− ν₁, γ2− ν₂, ν1+ m, ν2+ n; qi∞

by[9,1.45,1.46]

= LHS.

(47)

We make a new proof of the following theorem by fractional q-integration.

Theorem 12. [9, (7.50) p. 251], a q-analogue of [8, (1) p. 176].

2φ₁(α, β; γ|q; z) ∼= Γ_q(γ) Γ_q(β)Γ_q(γ − β)

Z 1 0

t^β−1(qt; q)_γ−β−1

(zt; q)_a d_q(t). (48) Proof.

LHS^by(36)= Γq(γ)

Γ_q(β)z^1−γD^β−γ_q

∞

X

k=0

hα; qi_k

h1; qi_kz^β+k−1

by[9,(7.27)p.247]

= Γq(γ)

Γ_q(β)z^1−γD^β−γ_q z^β−1 (z; q)_α

by(35)

= Γ_q(γ)z^1−γ Γq(β)Γq(γ − β)

Z z 0

t^β−1 (t; q)α

z^γ−β−1(^qt_z; q)∞

(_z^tq^γ−β; q)∞

dq(t)^by(33)= RHS.

(49)

Theorem 13. A q-analogue of Erd´elyi [8, (3) p. 176].

2φ₁(α, β; γ|q; z)

∼= Γ_q(γ) Γq(λ)Γq(γ − λ)

Z 1 0

t^λ−1 (qt; q)∞

(tq^γ−λq)∞ 2φ₁(α, β; λ|q; tz) d_q(t). (50) Proof.

LHS^by(36)= Γq(γ)

Γ_q(λ)z^1−γD^λ−γ_q

∞

X

k=0

hα, β; qi_k

hλ, 1; qi_kz^λ+k−1

=Γq(γ)

Γ_q(λ)z^1−γD^λ−γ_q 

z^λ−1 ₂φ₁(α, β; λ|q; z)

by(35)

= Γq(γ)z^1−γ Γ_q(λ)Γ_q(γ − λ)

Z z 0

z^γ−λ−1t^λ−1 (^qt_z; q)∞

(_z^tq^γ−λq)∞

2φ1(α, β; λ|q; t) dq(t)

by(33)

= RHS.

(51)

Theorem 14. A q-analogue of Erd´elyi [8, p. 182].

2φ₁(α, β; γ|q; z) ∼= Γq

 γ

µ, γ − λ

 Z 1

0

(qt; q)_γ−λ−1 (tz; q)α+β−λ

D^µ−λ_q,t t^µ−1 ₂φ₁(λ − α, λ − β; µ|q; tz)
d_q(t).

(52)

Proof. We find that LHS^by[9,(7.52)],(50)

= Γ_q

 γ

λ, γ − λ

 Z 1 0

t^λ−1(qt; q)_γ−λ−1

(tz; q)α+β−λ 2φ₁(λ − α, λ − β; λ|q; tz) d_q(t).

(53)

On the other hand, t^λ−1

Γ_q(λ) ²φ₁(λ − α, λ − β; λ|q; tz) = D^µ−λ_q,t

∞

X

k=0

hλ − α, λ − β; qi_k

h1; qi_kΓ_q(λ + k) t^µ+k−1z^k

= D^µ−λ_q,t  t^µ−1

Γq(µ) ²φ1(λ − α, λ − β; µ|q; tz)

 .

(54)

Theorem 15. A q-analogue of Erd´elyi [8, p. 184].

2φ₁(α, β; γ|q; z) ∼= Γq

 γ, µ λ, γ − λ, ν

 Z 1

0

t^λ−µ(qt; q)γ−λ−1D^ν−µ_q,t t^ν−1 3φ2(α, β, µ; λ, ν|q; tz)

dq(t).

(55)

Proof. We find that

t^µ−1 2φ1(α, β; λ|q; tz) = D^ν−µ_q,t

∞

X

k=0

hα, β; qi_kΓ_q(µ + k)

hλ, 1; qi_kΓq(ν + k)t^ν+k−1z^k

!

= D^ν−µ_q,t  Γq(µ)

Γ_q(ν)t^ν−1 3φ2(α, β, µ; λ, ν|q; tz)

 .

(56)

Finally, put this into formula (50).

The following formula, whose proof can be found in (80) for x = 0 and permutation of the parameters, corresponds to Rodriguez formula.

Theorem 16. A q-analogue of Koschmieder [20, (10.5)p.252] and Erd´elyi [8, (9) p. 178].

Almost a q-analogue of Kamp´e de F´eriet [19, p. 26].

2φ1(α, β; γ|q; z) = z^1−γΓq

 γ β

 D^β−γ_q,z

 z^β−1 (z; q)_α



. (57)

\section\label{Feld}

Definition 4. The first q-Lauricella function is

Φ⁽ⁿ⁾_A (a,~b; ~c|q; ~x) ≡X

~ m

ha; qi_mh~b; qi_m~~x^m~

h~c,~1; qi_m~ , (58)

where [10]

|x₁| ⊕_q· · · ⊕_q|x_n| < 1. (59)

The following formula, similar to [12], was not included there.

Theorem 17. (A q-analogue of Feldheim [14, (9) p. 244]).

Γq

 α, δ − α δ



Φ⁽ⁿ⁾_A (α, ~β; ~γ|q; ~x)

∼= Z 1

s=0

s^α−1(qs; q)_δ−α−1Φ⁽ⁿ⁾_A (δ, ~β; ~γ|q; s~x) d_q(s).

(60)

Proof. Compute the RHS:

RHS^by[9,6.54]

=

∞~

X

~ m=~0

hδ; qi_mh~β; qi_m~~x^m~

h~1, ~γ; qi_m~ (1 − q)

∞

X

k=0

q^k(α+m)h1 + k; qi_δ−α−1

by[9,6.8,6.10]

=

∞~

X

~ m=~0

hδ; qi_mh~β; qi_m~~x^m~

h~1, ~γ; qi_m~ (1 − q)

∞

X

k=0

q^k(α+m)hδ − α; qi_kh1; qi_∞ h1; qi_khδ − α; qi_∞

by[9,7.27]

=

∞~

X

~ m=~0

hδ; qi_mh~β; qi_m~~x^m~

h~1, ~γ; qi_m~ (1 − q) hm + δ, 1; qi_∞ hδ − α, α + m; qi∞

by[9,1.45,1.46]

= LHS.

(61)

Definition 5. Convergence regions for the following triple functions were given in [11].

The q-analogues of the Srivastava triple hypergeometric functions are H_A(a, b₁, b₂; c₁, c₂|q; x₁, x₂, x₃) ≡

∞

X

m,n,p=0

ha; qi_m+phb₁; qi_m+nhb₂; qi_n+p h1, c₁; qimh1; qi_nh1; qi_phc₂; qin+p

x^m₁ xⁿ₂x^p₃. (62)

H_B(a, b1, b2; c1, c2, c3|q; x₁, x2, x3) ≡

∞

X

m,n,p=0

ha; qi_m+phb₁; qi_m+nhb₂; qi_n+p

h1, c₁; qi_mh1, c₂; qi_nh1, c₃; qi_px^m₁ xⁿ₂x^p₃. (63)

H_C(a, b₁, b₂; c|q; x₁, x₂, x₃) ≡

∞

X

m,n,p=0

ha; qi_m+phb₁; qi_m+nhb₂; qi_n+p

h1; qi_mh1; qi_nh1; qi_phc; qi_m+n+px^m₁ xⁿ₂x^p₃. (64) Theorem 18. A q-integral representation of H_A. A q-analogue of [5, (2.1) p. 115].

H_A(a₁, a₂, a₃; c₁, c₂|q; x, y, z)

= Γ_q

 b

a1, b − a1

 Z 1 s=0

s^a1−1(qs; q)_b−a1−1H_A(b, a₂, a₃; c₁, c₂|q; xs, y, zs) d_q(s). (65)

Proof. Put

D ≡ Γq

 b

a1, b − a1



∞

X

m,n,p=0

hb; qi_m+pha₂; qi_m+nha₃; qi_n+p

h1, c₁; qi_mh1; qi_nh1; qi_phc₂; qi_n+px^myⁿz^p.

(66)

Then we have

RHS^by[9,6.54]

= D(1 − q)

∞

X

k=0

q^k(a1+m+p)h1 + k; qi_b−a1−1

by[9,6.8,6.10]

= D(1 − q)

∞

X

k=0

q^k(a1+m+p)hb − a₁; qi_kh1; qi_∞ h1; qi_khb − a₁; qi∞ by[9,7.27]

= D(1 − q) hb + n + p, 1; qi_∞ ha₁+ m + p, b − a₁; qi∞

by[9,1.45,1.46]

= LHS.

(67)

Theorem 19. A q-integral representation of HA. A q-analogue of [5, (2.2) p. 115].

H_A(a₁, a₂, a₃; c₁, c₂|q; x, y, z)

= Γ_q

 b

a2, b − a2

 Z 1 s=0

s^a2−1(qs; q)_b−a2−1H_A(a₁, b, a₃; c₁, c₂|q; xs, ys, z) d_q(s). (68) Theorem 20. A q-integral representation of H_A. A q-analogue of [5, (2.3) p. 115].

H_A(a₁, a₂, a₃; c₁, c₂|q; x, y, z)

= Γ_q

 b

a3, b − a3

 Z 1 s=0

s^a3−1(qs; q)_b−a3−1H_A(a₁, a₂, b; c₁, c₂|q; x, ys, zs) d_q(s). (69) Theorem 21. A q-integral representation of H_A. A q-analogue of [5, (2.4) p. 115].

H_A(a₁, a₂, a₃; c₁, c₂|q; x, y, z)

= Γ_q

 c₁ c1− b, b

 Z 1 s=0

s^b−1(qs; q)_c1−b−1H_A(a₁, a₂, a₃; b, c₂|q; xs, y, z) d_q(s). (70) Proof. Put

D ≡ Γ_q(c₁) Γq(c1− b)Γ_q(b)

∞

X

m,n,p=0

ha₁; qi_m+pha₂; qi_m+nha₃; qi_n+p h1, b; qi_mh1; qi_nh1; qi_phc₂; qin+p

x^myⁿz^p.

Then we have

RHS^by[9,6.54]

= D(1 − q)

∞

X

k=0

q^k(b+m)h1 + k; qi_c1−b−1

by[9,6.8,6.10]

= D(1 − q)

∞

X

k=0

q^k(b+m)hc₁− b; qi_kh1; qi_∞ h1; qi_khc₁− b; qi_∞

by[9,7.27]

= D(1 − q) hc₁+ m, 1; qi∞

hb + m, c₁− b; qi_∞

by[9,1.45,1.46]

= LHS.

(71)

Theorem 22. A q-integral representation of H_A. A q-analogue of [5, (2.5) p. 116].

H_A(a₁, a₂, a₃; c₁, c₂|q; x, y, z)

= Γq

 c₂ c2− b, b

 Z 1 s=0

s^b−1(qs; q)c2−b−1HA(a1, a2, a3; c1, b|q; x, ys, zs) dq(s). (72) Theorem 23. A q-integral representation of H_C. A q-analogue of [6, (2.1) p. 115].

H_C(a₁, a₂, a₃; c|q; x, y, z)

= Γq

 b

a1, b − a1

 Z 1 s=0

s^a1−1(qs; q)b−a1−1HC(b, a2, a3; c|q; xs, y, zs) dq(s). (73) Theorem 24. A q-integral representation of H_B. A q-analogue of [4, (3.1) p. 2757].

H_B(a₁, a₂, a₃; c₁, c₂, c₃|q; x, y, z)

= Γq

 a₁+ a₂ a1, a2

 ∞

X

m,n,p=0

ha₁+ a2; qi2m+n+pha₃; qin+p

h1, c₁; qimh1, c₂; qinh1, c₃; qip

x^m₁ xⁿ₂x^p₃ Z 1

s=0

s^a1+m+p−1(qs; q)_a2+m+n−1d_q(s).

(74)

Proof. We compute the right hand side:

Γq

 a1+ a2

a1, a2

 ∞

X

m,n,p=0

ha₁+ a2; qi2m+n+pha₃; qin+p

h1, c₁; qi_mh1, c₂; qi_nh1, c₃; qi_px^m₁ xⁿ₂x^p₃

×Γ_q

 a₁+ m + p, a₂+ m + n a1+ a2+ 2m + n + p



(75)

=

∞

X

m,n,p=0

ha₁+ a₂; qi_2m+n+pha₃; qi_n+p h1, c₁; qi_mh1, c₂; qi_nh1, c₃; qi_p

ha₁; qi_m+pha₂; qi_m+n

ha₁+ a₂; qi_2m+n+p x^m₁ xⁿ₂x^p₃ = LHS.

Definition 6. The fourth q-Lauricella function is defined by Φ⁽ⁿ⁾_D (a,~b; c|q; ~x) ≡X

~ m

ha; qi_mh~b; qi_m~~x^m~

hc; qi_mh~1; qi_m~ , max(|x₁|, . . . , |x_n|) < 1. (76) Theorem 25. A q-analogue of Srivastava and Manocha [23, p.289 (17)].

D^λ−µ_q,z

 z^λ−1

(az, q)_α(bz, q)_β(cz, q)_γ



= Γ_q(λ)

Γq(µ)z^µ−1Φ⁽³⁾_D (λ, α, β, γ; µ|q; az, bz, cz).

(77)

Proof.

LHS = D^λ−µ_q,z



z^λ−1

∞

X

m,n,p=0

hα; qi_mhβ; qi_nhγ; qi_p

h1; qi_mh1; qi_nh1; qi_p a^mbⁿc^pz^m+n+p





by[9,8.118]

=

z^µ−1Xhα; qi_mhβ; qi_nhγ; qi_p(az)^m(bz)ⁿ(cz)^p h1; qi_mh1; qi_nh1; qi_p Γq

 λ + m + n + p µ + m + n + p



by[9,1.45,1.46]

= RHS.

(78)

The following operator formula is a generalization of (57).

Theorem 26. A q-analogue of Srivastava and Manocha [23, p.289 (18)].

D^λ−µ_q,y



y^λ−1 1

(y; q)α 2φ1(α, β; γ|q; x||−; yq^α)



= Γ_q(λ)

Γq(µ)y^µ−1Φ2(α, β; λ; γ, µ|q; x, y).

(79)

Proof.

LHS = D^λ−µ_q,y





∞

X

m,n=0

hα; qi_m+nhβ; qi_m

hγ, 1; qi_mh1; qi_n x^my^λ+n−1



 (80)

by[9,8.118]

=

∞

X

m,n=0

hα; qi_m+nhβ; qi_mx^m

hγ, 1; qi_mh1; qi_n y^µ+n−1Γ_q

 λ + n µ + n

by[9,1.45,1.46]

= RHS.

Definition 7. Assume that ~m ≡ (m1, . . . , mn), m ≡ m1+ . . . + mn and a ∈ R^?. The vector q-multinomial-coefficient _m~^a
?

q [13] is defined by the symmetric expression

 a

~ m

? q

≡ h−a; qi_m(−1)^mq⁻(^m~₂)^+am h1; qi_m1h1; qi_m2. . . h1; qimn

. (81)