Some new Formulas involving <span class="mathjax-formula">$\Gamma _q$</span>Functions

Some New Formulas Involving G

Functions.

THOMASERNST(*)

ABSTRACT- In a recent paper we found some new results forq-functions of many variables with the aid of theGqfunction. The Heine notation reminding of the hypergeometric case was used throughout, and some relations between Gq

functions were presented. This paper aims at giving the promised longer ex- position ofGq- revealing also the connection between this and the Jacobi-theta functions which appearin context. We will give a slightly generalized definition of the Heine series with more general tilde operators. 4q-summation formulas of Andrews will be given in the new notation. The close affinity to q-binomial coefficient formulas will be stressed by expressing the finiteq-hypergeometric formulas, the canonical form, in two ways. Two furtherq-analogues of Kummer's

2F1( 1) formula will be given. An ancientq-analogue of the Eulerreflection formula will be used for the proof of a special case of the Bailey-Daum sum- mation formula, conjectured in the previous paper.

Multiple extensions of Gauss' formula will be given by a similar technique. All this will explain the utility of the Heine notation.

1. Introduction.

In this section we present the necessary definitions together with useful related formulas. The notation from [13] will be used whenever possible. The umbral calculus from [15] will play a significant role in the definition of the q-hypergeometric series. The Jackson G_q function fits nicely togetherwith theq-shifted facorial. In Section two, 2q-summation formulas of Andrews are given in terms ofGqfunctions, and 2q-summation formulas of Andrews will be proved by Watson's [46] transformation for- mula fora terminating very-well-poised ₈f₇ series. In Section three two furtherq-analogues of Kummer's ₂F₁( 1) formula will be given. We will prove a special case of the Bailey-Daum summation formula, conjectured in

(*) Indirizzo dell'A.: Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 Uppsala, Sweden; E-mail: [email protected]

2000 Mathematics Subject Classification: Primary 33D05; Secondary 33D70.

the previous paper [13], by using Jacobi-theta functions. In Section four multiple extensions of Gauss' formula will be given by a similar technique.

1.1 ±Tilde operators.

For preliminary definitions the reader is referred to the paper [13]. In that paperwe defined aq-shifted factorial and a tilde operator.

DEFINITION1. The operator e:C

Z7!C Z is defined by

a7!a‡ pi logq: …1†

By (1) it follows that

ha;gqi_nˆⁿY¹

mˆ0

(1‡q^a‡m);

…2†

It turns out that in certain formulas it is not easy to cope with the caseaˆ0:

That's why we define

he0;qi_n he1;qi_{n 1}: …3†

This means that ifaˆ0 we skip the first factor 1‡1, which is not really aq- shifted facorial, and just compute the lastn 1 factors. The reason is the affinity with the factor1 1, which causes trouble in denominators.

Because

h1 gn;qi_n ˆh0;eqi_nq … †ⁿ² ; …4†

we also define

h1 gn;qi_n h1 gn;qi_{n 1}: …5†

This formula was already used by Watson [47, p. 64].

To be able to treat a general root of unity, it is desirable to generalize this operator in the following way. In equations (6) to (16) we assume that (m;l)ˆ1.

DEFINITION2. The operator e

ml :C Z7!C

Z is defined by

a7!a‡2pim llogq: …6†

This means

h^mf_la;qi_nˆY^{n 1}

mˆ0

(1 e^2pimlq^a‡m);

…7†

Furthermore we define

g

mlha;qi_n hf^m_la;qi_n: …8†

We will also need another generalization of the tilde operator.

h_kea;qi_nY^{n 1}

mˆ0

X^{k 1}

iˆ0

q^i(a‡m)

: …9†

h₂ea;qi_n hea;qi_n: …10†

h₁ea;qi_n1:

…11†

kha;gqi_n h_kea;qi_n: …12†

The following simple rules follow from (6). Some of them were pre- viously known in othernotation from [17].

THEOREM1.1.

f

mlab^m_l(agb)mod 2pi logq; …13†

X^l

kˆ1

g

1la_kX^l

kˆ1

a_k mod 2pi logq; …14†

m

l eamg

2lam

l mod 2pi logq; …15†

QE(^me_la)ˆQE(a)e^2piml ; …16†

where the second equation is a consequence of the fact that we work mod 2pi

logq. Furthermore, THEOREM1.2.

hea;q²i_nˆ h¹₄ea;³₄ea;qi_n: …17†

ha;q^pi_n ˆY^{p 1}

kˆ0

h^k_pea;qi_n; where p is an odd prime:

…18†

ha;q^ki_n ˆ ha;qi_nkha;gqi_n: …19†

This leads to the following q-analogue of [36, p. 22, (2)].

THEOREM1.3.

ha;qi_knˆY^{k 1}

mˆ0

Da‡m k ;qE

nkD ga‡m k ;qE

n: …20†

1.2 ±Heine's series

The q-hypergeometric series was developed by Heine 1846 [24] as a generalization of the hypergeometric series.

DEFINITION 3. Generalizing Heine's series, we shall define a q-hy- pergeometric series by (compare [17, p. 4], [22, p. 345]):

p‡p⁰f_r‡r0(a^1;. . .;a^p; ^b1;. . .;b^rjq;zks1;. . .;sp⁰;t1;. . .;tr⁰)

p‡p⁰f_r‡r0

a^₁;. . .;a^_p

b^₁;. . .;b^_rjq;zks1;. . .;sp⁰

t₁;. . .;t_r⁰

" #

ˆ

ˆX¹

nˆ0

h^a1;qi_n. . .h^ap;qi_n

h1;qi_nh^b₁;qi_n. . .h^b_r;qi_nh( 1)ⁿq… †ⁿ² i_1‡r‡r⁰ _{p p}⁰

zⁿY^p0

kˆ1

(s_k;q)_nY^r0

kˆ1

(t_k;q)_n¹; …21†

whereq6ˆ0 whenp‡p⁰>r‡r⁰‡1;and

baˆ a;

ea;

f

mla;

kea 8>

>>

>>

<

>>

>>

>: …22†

In a few cases the parameter^ain (21) will be the real plus infinity (0<jqj<1). They correspond to multiplication by 1. If we want to be formal, we could introduce a symbol1_H;with property

h1_H;qi_n ˆ h1_H‡a;qi_nˆ1; a2C; 0<jqj<1:

…23†

The symbol1Hcorresponds to parameter 0 in [17, p. 4]. We will denote1H

by1in the rest of the paper.

The terms to the left ofjin (21) are thought to be exponents, they are periodic mod 2pi

logq.

The first term to the right ofjin (21) is the base, and the second is a letter in the spirit of the umbral calculus in [15]. The terms to the right ofk in (21) are Watson q-shifted factorials [17]. There will be a certain re- dundance in notation here, but this is no problem.

1.3 ±TheGqfunction

In the same way as theGfunction plays a basic role in complex analysis, theG_qfunction is fundamental forq-calculus. During the last years there has been an increasing interest in the Gq function. Howeverthere are certain restrictions in our knowledge of this function as the following ex- position shows. TheG_qfunction is defined in the unit disk 0<jqj<1 by

DEFINITION4.

Gq(x)ˆh1;qi₁

hx;qi₁(1 q)^{1 x}: …24†

Here we deviate from the usual conventionq<1, because we want to work with meromorphic functions of several variables. The simple poles are located atxˆ n 2kpi

logq; n;k2N.

Heine, Ashton and Daum [11] used anotherfunction without the factor (1 q)^{1 x}, which they called the Heine V-function. The main difference between the two functions is thatVhas zeros, in contrast to theGqfunction which has no zeros, and therefore 1

Gqis entire. Ashton [5] in his thesis su- pervised by Lindemann, showed its connection to elliptic functions. Daum [11] tried to find all the basic analogues of Thomae's3F2 transformation formula [42, eq. 11], using a notation analogous to that used by Whipple [48], and by essentially replacing theG_qfunction by the HeineV-function.

Daum's work was continued when in 1987 Beyer - Louck - Stein [9] and in 1992 Srinivasa Rao - Van der Jeugt - Raynal - Jagannathan - Rajeswari [38]

showed that certain two-term transformation formulas between hypergeo- metric series easily can be described by means of invariance groups. In other words, they explained Whipple's [48] discovery in group language. In 1999 Van derJeugt - Srinivasa Rao [43] foundq-analogues of these results.

Daum [11] concludes his thesis by sayingIt is hoped, however, that the use of the modified HeineV-function, will serve to emphasize the analogy between hypergeometric series and q-hypergeometric series and simplify the notation generally.

Let's now return to theG_q function: To save space, the following no- tation forquotients ofGq functions will often be used.

DEFINITION5. The generalizedGqfunction is given by G_q a₁;. . .;a_p

b₁;. . .;b_r

G_q(a₁). . .G_q(a_p) Gq(b1). . .Gq(br): …25†

DEFINITION6. This generalizedGqfunction is called balanced if X^p

kˆ1

a_k ˆX^r

kˆ1

b_k; pˆr:

…26†

DEFINITION7. A quotient of infiniteq-shifted factorials Q^p

kˆ1ha_k;qi₁ Q^r

kˆ1hb_k;qi₁ …27†

is called balanced if

X^p

kˆ1

a_k ˆX^r

kˆ1

b_k; pˆr:

…28†

These definitions are easily seen to be equivalent.

Balanced quotients ofGq's occurin many equations ofq-calculus. To see why this is the case, we need to go back 100 years to the work of Mellin [31].

Mellin discovered that every hypergeometric function can be written as a function of generalizedG functions. In the same way, every q-hypergeo- metric function can be written as a function of generalizedG_qfunctions [23, p. 258]. In the presence of a balanced quotient of Gq's, limits when the parameters tend to 1usually pose no problem. It would therefore be agreeable if we could find bounds for quotients of balancedGq's. The first step in this direction was made by Ismail and Muldoon [25, p. 320]. This was generalized by Alzer [1].

The following lemmas are just a few examples ofG_q formulas.

LEMMA1.4. A relation betweenGq functions with different bases.

G_q²(x)

G_q(x) ˆhe1;qi₁

hex;qi₁(1‡q)^{1 x}(1‡q)^{1 x}he1;qi_{x 1}: …29†

LEMMA1.5.

D1‡a

2 ;1 b‡a 2;q²E h1‡a b;q²i₁ 1ˆ

G_q 1‡a b;1‡a 2 1‡a;1‡a

2 b 2

64

3 75

D g1‡a 2 b;qE g 1

D1‡a

2;1‡ga b;qi₁ : …30†

PROOF. Use the Bailey-Daum theorem. p

We are going to find q-analogues of hypergeometric summation for- mulas in this paper. One example is the Whipple formula [48] for a ter- minating series, which was first given by Watson.

…31† 3F2 a; n;c

2;1‡a n 2 ;c;1

ˆG

1‡a n 2 ;1‡c

2 ;1‡c a‡n

2 ;1

2 1‡a

2 ;1 n

2 ;1 a‡c

2 ;1‡c‡n 2 2

66 4

3 77 5

2. Fourq-summation formulas of Andrews.

In recent years, combinatorics has developed quickly. Combinatorial identities can be expressed either as hypergeometric formulas - the ca-

nonical form-, or as binomial coefficient identities. Combinatorial results are often expressed asq-formulas. Here we have a similar duality:q-hy- pergeometric formulas - the canonical form-, or q-binomial coefficient identities. Nowq-binomial coefficient computations are easier to handle by hand, orby computer. Some mathematicians, like forexample Catalan, Bateman and Gould, have tried (in vain) to make systematic treatments of binomial coefficient identities. Nonetheless, q-hypergeometric formulas and q-binomial coefficient identities belong together. Therefore we re- present many theorems of this chapter both ways.

As a basis forthe following calculations, we are going to use 4 q-for- mulas by Andrews. The first is

THEOREM2.1. Andrews's q-analogue of Kummer's formula[28, (2), p. 134]from[2, 1.8 p. 526].

…32†

2f₂

a;b 1‡a‡b

2 ;1‡ga‡b 2

jq; q 2

4

3 5ˆGq

1‡a‡b 2 ;1

2 1‡b

2 ;1‡a 2 2

66 4

3 77 5

g1‡b 2 ;q

a2

e1 2;q

a2

G_q²

1‡a‡b 2 ;1

2 1‡b

2 ;1‡a 2 2

66 4

3 77 5:

PROOF. By [2, 1.8 p. 526]

LHS^{by (30)}ˆ Gq

1‡a‡b 2 ;1‡a

2 1‡a;1‡b

2 2

66 4

3 77 5

g1‡b 2 ;e1;q

g1‡a‡b 1

2 ; g

1‡a 2;q

1 by (29)ˆ

G_q

1‡a‡b 2 1‡b

2 ;1‡a 2

66 4

3 77

5G_q² 1‡a 2 2 4

3 5(1‡q)^a2

g1‡b 2 ;q

g1‡a‡b 1

2 ;q

1

ˆ …33†

G_q

1‡a‡b 2 1‡b

2 2 66 4

3 77 5G_q2

1 2 1‡a

2 2 66 4

3 77

5(1‡q) ^a2

g1‡b 2 ;q

g1‡a‡b 1

2 ;q

1

by (29)ˆ RHS;

where we have used theq-analogue of the Legendre duplication formula in

the penultimate step. p

REMARK 1. The name Gauss second summation theorem for Kummer's formula [28, (2), p. 134] was coined by Slater. This formula has recently received a lot of attention. We will generalize the q- analogue later.

Ifbis a negative integer, (32) may be reformulated in the form

THEOREM2.2.

2f₂

a; 2N 1‡a 2N

2 ;1‡ga 2N 2

jq; q 2

4

3 5

X^2N

kˆ0

2N k

q q²… †^{k2‡k(1 2N)} ( 1)^kha;qi_k D1‡a 2N

2 ;q²E

k

ˆ

D1 2;q²E

Nq ^Na D1 a

2 ;q²E

N

D1 2N 2 ;q²E D1‡a 2N N

2 ;q²E

N

: …34†

2f₂

a; N 1‡a N

2 ;1‡ga N 2

jq; q 2

4

3 5

X^N

kˆ0

N

k _q q²… †^{k2 ‡k(1 N)} ( 1)^kha;qi_k D1‡a N

2 ;q²E

k

ˆ0;N odd:

…35†

The following corollary was influenced by [27]. The proof is the same.

This idea to use the contiguity relations goes back to Kummer [28, p.

134-36].

COROLLARY2.3.

2f₂

a;b 2‡a‡b

2 ;2‡ga‡b 2

jq; q 2

4

3

5ˆq^a(1 q^b) q^a q^b Gq

2‡a‡b 2 ;1 2‡b 2

2 ;1‡a 2 2

66 4

3 77 5

g2‡b 2 ;q

a2

e1 2;q

a2

q^b(1 q^a) q^a q^b Gq

2‡a‡b 2 ;1 1‡b 2

2 ;2‡a 2 2

66 4

3 77 5

g1‡b 2 ;q

a‡12

e1 2;q

a‡12

: …36†

2f₂

a;b 3‡a‡b

2 ;3‡ga‡b 2

jq; q 2

4

3 5ˆ

q^a(1 q^b) q^a q^b

q^a(1 q^b‡1) q^a q^b‡1 G_q

3‡a‡b 2 ;1

2 3‡b

2 ;1‡a 2 2

66 4

3 77 5

g3‡b 2 ;q

a2

e1 2;q

a2

2 66 66 4

q^b‡1(1 q^a) q^a q^b‡1 Gq

3‡a‡b 2 ;1

2 2‡b

2 ;2‡a 2 2

66 4

3 77 5

g2‡b 2 ;q

a‡12

e1 2;q

a‡12

3 77 77 5

q^b(1 q^a) q^a q^b

q^a‡1(1 q^b) q^a‡1 q^b Gq

3‡a‡b 2 ;1

2 2‡b

2 ;2‡a 2 2

66 4

3 77 5

g2‡b 2 ;q

a‡12

e1 2;q

a‡12

2 66 66 4

q^b(1 q^a‡1) q^a‡1 q^b G_q

3‡a‡b 2 ;1 1‡b 2

2 ;3‡a 2 2

66 4

3 77 5

g1‡b 2 ;q

a‡22

e1 2;q

a‡22

3 77 77 5: …37†

THEOREM2.4. Andrews's q-analogue of Kummer's formula[28, p. 134]

from[2, 1.8 p. 526]and[39, (15), p. 9].

2f₂ a;1 a c;e1 jq; q^c

" #

ˆGq

c 2;1‡c

2 1‡c a

2 ;a‡c 2 2

66 4

3 77 5

g1‡c a 2 ;q

a2

ec 2;q

a2

G_q² c 2;1‡c 1‡c a2

2 ;a‡c 2 2

66 4

3 77 5: …38†

PROOF. By [2, 1.9 p. 526]

LHSˆ

D1‡c a 2 ;a‡c

2 ;q²

hc;qi₁ 1ˆ D1‡c a

2 ;1‡gc a 2 ;a‡c

2 ;ag‡c 2 ;q

1

1‡c 2 ;1g‡c

2 ;c 2;ec

2;q

1

ˆRHS:

…39†

p Ifais a negative integer, (38) may be reformulated in the form THEOREM2.5.

2f₂ 2N;1‡2N c;e1 jq; q^c

" # X^2N

kˆ0

2N k

q ( 1)^kq²… †^{k2 ‡k(c 2N)}h1‡2N;qi_k hc;e1;qi_k ˆ

Dc 2N 2 ;q²E D1‡c N

2 ;q²E

N

: …40†

2f₂ N;1‡N c;e1 jq; q^c

" #

ˆ Dc N

2 ;q²E

1‡N2

Dc 2;q²E

1‡N2

;N odd:

…41†

We have expressed two of Andrew's formulas by theGq function, and also expressed the corresponding finite sums by quotients of q-shifted factorials. We will continue with generalizations of these two formulas.

We need a lemma forthe following proof.

LEMMA2.6. A q-analogue of the Legendre duplication formula for the G-function written in q-shifted factorial form is

h1 n;q²iⁿ₂ h1 gn;qi_{n 1}D1

2;q²E

n2

ˆ( 1)ⁿ² QE n² 4

;neven:

…42†

THEOREM2.7. Compare[18, (II 17), p. 355].A q-analogue of[45], [8, (1.2), p. 237].

4f₃

c 2;ec

2;a; 2N 2N‡1‡a

2 ; 2Ng‡1‡a

2 ;c

jq;q 2

66 64

3 77 75ˆ

D1

2;1‡c a 2 ;q²E D1 a N

2 ;1‡c 2 ;q²E

N

: …43†

4f₃

c 2;ec

2;a; N N‡1‡a

2 ; Ng‡1‡a

2 ;c

jq;q 2

66 4

3 77

5ˆ0; N odd:

…44†

PROOF. The proof of (44) is relegated to the next proof, i.e. (54). We will use the method in Andrews [3, p. 334]. For convenience, we will denote the

LHS of (43)₄f₃

c 2;ec

2;a; n n‡1‡a

2 ; n‡g1‡a 2 ;c

jq;q 2

66 4

3 77 5:

By Watson's [46] transformation formula for a terminating very-well- poised8f₇ series, denoting

(a)(a;b;c;d;1‡1 2a; g

1‡1

2a;e; n);

…45†

(b)(1‡a b;1‡a c;1‡a d;1‡a e;1 2a;1f

2a;1‡a‡n);

…46†

this formula takes the following form

8f₇ (a)

(b)jq;q2a‡2‡n b c d e

ˆh1‡a;1‡a d e;qi_n h1‡a d;1‡a e;qi_n

4f₃ d;e;1‡a b c; n

1‡a b;1‡a c;d‡e n ajq;q

: …47†

Now make the substitution a!gn;b! n‡1 a

2 ;c! n‡g1 a 2 ;d!c

2;e!ec 2; …48†

to obtain

8f₇ (a⁰)

(b⁰)jq;q^{1‡a c}

g

n‡1 c

2; n‡1 c 2;q

gn‡1; n‡1 c;q n

n

ˆ

4f₃

c 2;ec

2;a; n n‡1‡a

2 ; n‡g1‡a 2 ;c

jq;q 2

66 4

3 77 5: …49†

where

(a⁰) gn; n‡1 a

2 ; n‡g1 a 2 ;c

2;ec 2;₁ g

41 1

2n;₃ g

41 1

2n; n

!

; …50†

(b⁰)( n‡1‡a

2 ; n‡g1‡a

2 ;1 n c

2; g

1 n c

2;1g

4 n

2 ;3g

4 n

2 ;e1);

…51†

Now

hea;q²i_nˆ e¹₄a;³₄ea;q

n: …52†

By theq-Dixon theorem we have, assumingneven

4f₃

c 2;ec

2;a; n n‡1‡a

2 ; n‡g1‡a 2 ;c

jq;q 2

66 4

3 77 5ˆ

4f₃

n‡1 a

2 ; g

1 n 2;c

2; n 1 n‡a

2 ;1 n c

2; gn 2

jq²;q^{1‡a c} 2

66 4

3 77 5

g n‡1 c

2; n‡1 c 2;q

gn‡1; n‡1 c;q n

n

ˆ

g n‡1 c

2; n‡1 c 2;q

gn‡1; n‡1 c;q n

n

1 n;1 n‡a c 2 ;q²

n2

n‡1 c

2;1 n‡a 2 ;q²

n2 by (42)ˆ

D1

2;1‡c a 2 ;q²E

n2

D1 a 2 ;1‡c

2 ;q²E

n2

: …53†

p

Equation (43) may be reformulated in the form THEOREM2.8. Compare[3, p. 334].

4f₃

c 2;ec

2;a; n n‡1‡a

2 ; n‡g1‡a 2 ;c

jq;q 2

66 4

3 77 5

ˆ e1;1 c;q g1 n; n‡1 1c;q

1

D 2n‡2

2 ;1‡a

2 ; n‡2 c

2 ;a n‡1 c 2 ;q²

D2 c 1

2 ;1‡a n

2 ; n‡2

2 ;a‡1 c 2 ;q²

1

: …54†

PROOF. LHSˆ4f₃

n‡1 a

2 ; g

1 n 2;c

2; n

1 n‡a

2 ;1 n c

2; gn 2

jq²;q^{1‡a c} 2

66 4

3 77 5

g n‡1 c

2; n‡1 c 2;q

gn‡1; n‡1 c;q n

n

ˆ

e1;1 c;q

1

Da‡1

2 ; n‡1; n‡2 c

2 ; n‡a‡1 c 2 ;q²

1

gn‡1; n‡1 c;q

1

Da n‡1 2 ; n‡2

2 ;2 c

2 ;a‡1 c 2 ;q²

1

ˆ

e1;q

1

Da‡1 2 ; n‡1

2 ;1 c

2 ; n‡a‡1 c 2 ;q²

Da n‡1 1

2 ; n‡1 c

2 ;a‡1 c 2 ;q²

1

ˆ e1;1 c;q g1 n; n‡1 1c;q

1

D 2n‡2 2 ;1‡a

2 ; n‡2 c

2 ;a n‡1 c 2 ;q²

D2 c 1

2 ;1‡a n 2 ; n‡2

2 ;a‡1 c 2 ;q²

1

: …55†

p REMARK2. The equivalent expression

Da‡1

2 ; n‡1 2 ;1 c

2 ; n‡a‡1 c

2 ;q²

D1 1

2;a n‡1

2 ; n‡1 c

2 ;a‡1 c 2 ;q²

1

…56†

forthe RHS of (54) was given in [21, 1.4.]. The simple proof uses the Euler formula [16, p. 271]

e1;q

1

1 2;q²

1ˆ1:

…57†

This was the beginning of the investigation that led to the current paper.

Now letc! 1in (54) and (43) to arrive at (32) and (34).

COROLLARY2.9.

3f₂

a; 2N;1 1‡a 2N

2 ;1‡ga 2N 2

jq;q 2

4

3 5ˆ

D1 2;q²E D1 a N

2 ;q²E

N

: …58†

PROOF. Letc! 1in (43). p

COROLLARY2.10.

3f₂ c 2;ec

2; 2N

c;1 jq; q

" #

X^2N

kˆ0

2N k

qq… †^{k2 ‡k(1 2N)}( 1)^kDc 2;q²E hc;qi_k k

ˆ D1

2;q²E

N q^Nc D1‡c

2 ;q²E

N

: …59†

3f₂ c 2;ec

2; N c;1 jq;q

" #

X^N

kˆ0

N

k _q q… †^k2‡k(1 N)( 1)^kDc 2;q²E

hc;qi_k kˆ0;N odd:

…60†

PROOF. Leta! 1in (43) and (44). p

COROLLARY2.11. Compare[18, ex. 3.4, p. 101].

3f₁ c 2;ec

2; 2N

c

jq; q^2N 2

4

3 5X^2N

kˆ0

2N k

q

( 1)^kDc 2;q²E hc;qi_k k

ˆ D1

2;q²E D1‡c N

2 ;q²E

N

: …61†

3f₁ c 2;ec

2; N c

jq; q^N 2

4

3 5X^N

kˆ0

N k _q

( 1)^kDc 2;q²E

hc;qi_k kˆ0;Nodd:

…62†

PROOF. Leta! 1in (43) and (44). p

THEOREM2.12. [3, p. 333]

4f₃ c 2;ec

2;1‡n; n c‡1 e;e1;e

jq;q 2

64

3 75ˆ

ee;e c;q 1

g

e n; n‡e c;q

1

De‡1‡n

2 ; n‡e‡1 c

2 ; n‡e;q² D n‡e‡1 1

2 ;n‡1‡e c 2 ;e;q²

1

: …63†

PROOF. We will again use Watson's 1929 [46] transformation formula fora terminating very-well-poised8f₇series. Now make the substitution

a! n‡ge 1;b! n‡e 1;c!gn;d!c 2;e!ec

2: …64†

to obtain

8f₇ (a⁰)

(b⁰)jq;q^{e‡1 a c}

g

n‡e c

2; n‡e c 2;q

ng‡e; n‡e c;q n

n

ˆ

4f₃ c 2;ec

2;1‡n; n e;e1;c‡1 e

jq;q 2

64

3 75:

…65†

where

(a⁰) fn; n‡e 1; ng‡e 1;c 2;ec

2;₁ g

41‡e n

2 ;₃ g

41‡e n

2 ; n

!

; …66†

(b⁰)(e;ee;e n c

2; g

e n c

2;₁ g

4 n‡e 1

2 ;₃ g

4 n‡e 1

2 ;e1):

…67†

Now

hea;q²i_nˆ e¹₄a;³₄ea;q

n: …68†

And we have

4f₃ c 2;ec

2;1‡n; n e;e1;c‡1 e

jq;q 2

4

3 5ˆ

4f₃ n 1‡e;1 gn‡e 2 ;c

2; n e;e n c

2; n‡ge 1 2

jq²;q^{e‡1‡n c} 2

66 64

3 77 75

g n‡e c

2; n‡e c 2;q

n

gn‡e; n‡e c;qi_n

ˆ

hee;e c;qi₁Dn‡e‡1

2 ; n‡e; n‡e‡1 c 2 ;q²

gn‡e; n‡e c;q 1

1

e; n‡e‡1

2 ;n‡e‡1 c 2 ;q²

1

ˆ hee;e c;qi₁ g

e n; n‡e c;q

1

2n‡2e c

2 ;e‡1‡n

2 ; n‡e‡1 c 2 ;2e c

2 ; n‡e;q² 1

2e c

2 ; 2n‡2e c

2 ; n‡e‡1

2 ;n‡1‡e c 2 ;e;q²

1

ˆRHS:

…69†

p Equation (63) may be reformulated in the form

THEOREM2.13. A q-analogue of the Whipple formula[48, p. 114], [8, (1.3), p. 237].

4f₃ c 2;ec

2;1‡2N; 2N c‡1 e;e1;e

jq;q 2

4

3 5

X^2N

kˆ0

2N k

q q… †^{k2 ‡k(1 2N)}( 1)^kh1‡2N;qi_kDc 2;q²E

k

hc‡1 e;e1;e;qi_k

ˆ( 1)^N QE(N²‡N‡Nc Ne)De 2N

2 ;e‡1 c 2 ;q²E De‡1 N

2 ;2‡c e 2 ;q²E

N

: …70†

4f₃ c 2;ec

2;2‡2N; 2N 1 c‡1 e;e1;e

jq;q 2

4

3 5

^2N‡1X

kˆ0

2N‡1 k

q q… †^{k2 2kN}( 1)^kh2‡2N;qi_kDc 2;q²E

k

hc‡1 e;e1;e;qi_k

ˆ( 1)^N‡1 QE((N‡1)²‡(c e)(N‡1))De 2N 1 2 ;e c

2 ;q²E De N‡1

2;1‡c e 2 ;q²E

N‡1

: …71†

Now letc! 1in (70) and (71) to arrive at (40) and (41).

COROLLARY2.14. Anotherq-analogue of Kummers formula.

3f₂ 2N;1‡2N;1 c;e1 jq;q

X^2N

kˆ0

2N k

q ( 1)^kq… †^{k2‡k 2Nk}h1‡2N;qi_k hc;e1;qi_k ˆ

Dc 2N 2 ;q²E D1‡c N

2 ;q²E

N

q^2N2‡N: …72†

3f₂ N;1‡N;1 c;e1 jq;q

X^N

kˆ0

N

k _q ( 1)^kq… †^{k2‡k Nk}h1‡N;qi_k hc;e1;qi_k

ˆ Dc N

2 ;q²E

1‡N2

Dc 2;q²E

1‡N2

q… †^N‡12 ;N odd:

…73†

PROOF. Letc! 1in (70) and (71). p

3. Kummer's 2F1( 1)formula and Jacobi's theta function.

In [13] we used aq-analogue of the Dixon-Schafheitlin theorem

4f₃ a;b;c; g 1‡1 2a 1‡a b;1‡a c;1f

2a

jq;q^{1‡a2 b c} 2

66 64

3 77 75ˆ

ˆG_q

1‡a b;1‡a c;1‡a 2;1‡a

2 b c

1‡a;1‡a b c;1‡a

2 b;1‡a 2 c 2

66 4

3 77 5 …74†

to prove a q-analogue of Kummer's ₂F₁( 1) formula [28], the Bailey- Daum summation formula,

…75† 2f₁(a;b;1‡a bjq; q^{1 b})ˆGq

1‡a b;1‡a 2 1‡a;1‡a

2 b 2

4

3 5

1‡ga 2 b;e1;q

g 1

1‡a

2;1gb;q

1

;

where 1‡a b6ˆ0; 1; 2. . ..

We can easily find two related formulas.

THEOREM3.1. A second q-analogue of Kummer's 2F1( 1)formula.

3f₃

a;b; g 1‡a

2;1‡a b;ea

2;1jq;q^{1‡a2 b}

ˆGq

1‡a b;1‡a 2 1‡a;1‡a

2 b 2

4

3 5; …76†

where1‡a b6ˆ0; 1; 2. . ..

PROOF. Letc! 1in (74). This proof is aq-analogue of [7, p. 13]. p THEOREM3.2. A third q-analogue of Kummer's 2F1( 1)formula.

4f₂

a;b; g 1‡a

2;1;1‡a b;ea

2jq;q ^{a2 b}

ˆG_q 1‡a b;1‡a 2 1‡a;1‡a

2 b 2

4

3 5q^ab2; …77†

where1‡a b6ˆ0; 1; 2. . ..

PROOF. Letc! ‡ 1in (74). This proof is aq-analogue of [4, p. 144].

p

In [13] we found the following special case of (75)

2f₁( 2N;b;1 2N bjq; q^{1 b}) X^2N

kˆ0

2N k

q q… †^k2‡k(1 2N b) hb;qi_k h1 2N b;qi_k 2X^{N 1}

kˆ0

2N k

q( 1)^k hb;qi_k

h2N‡b k;qi_k‡ 2N N

q( 1)^N hb;qi_N hN‡b;qi_Nˆ heb;qi_N

hN‡b;qi_N D1

2;q²E

N: …78†

2f₁( N;b;1 N bjq; q^{1 b})ˆ0;Nodd:

…79†

The aim of the rest of this section is to give a proof of (78).

The proof forqˆ1 [35, p. 43] uses the Euler reflection formula G(x)G(1 x)ˆ p

sinpx: …80†

The following crucial formula has an interesting history. It was known in the literature twice before it was given in English in 2001. It first ap- peared 1873 in a book by Thomae about among other things theta func- tions. Then it appeared in an Italian paper 1923 by Pia Nalli. The inter- esting thing is that both Thomae and Pia Nalli used notations fortheta functions which are clearly different from the modern notation. The Tho- mae notation is reminiscent of Riemann theta functions and the Pia Nalli notation was probably influenced by nineteenth-century Italian books about elliptic functions. It is also rumoured that Andrews and Askey have known this formula for many years.

THEOREM3.3. The q-analogue of(80)is[41, p. 183 (168a)], [6, p. 1326], [32, p. 338]

G_q(x)G_q(1 x)ˆ i q¹⁼⁸(1 q)(h1;qi₁)³ q^x=2u₁

ix

2 logq;q¹⁼² ; …81†

where the first Jacobi theta function is given by u₁(z;q)ˆ2X¹

nˆ0

( 1)ⁿQE n‡1

2 ₂

sin (2n‡1)z:

…82†

This function has period2pand quasiperiod i 2 logq.

For the proof of (78) we need two further lemmata.

LEMMA3.4. Another q-analogue of the Legendre duplication formula written in q-shifted factorial form is

hN;qi_N h1 N;gqi_{N 1}D1

2;q²E

N

ˆQE N 2

: …83†

LEMMA3.5.

u₁ iNlogq;q¹⁼² u₁

iN

2 logq;q¹⁼²

ˆ( 1)^Nq ^(N=2):

…84†

PROOF. Use the quasiperiodicity ofu₁. p

PROOF. If we use the same method of proof as in [35, p. 43], but use (81) instead of (80), we arrive at

2f₁( 2N;b;1 2N bjq; q^{1 b})ˆ

1 gN b;e1;q g1 N;1gb;q1 1

h1 N b;N;qi₁

h1 2N b;2N;qi₁QEN 2

u1 iNlogq;q¹⁼² u1

iN

2 logq;q¹⁼² ˆ

1 gN b;N;qi_N g1 N;1 2N b;q

N

QE N

2

u₁ iNlogq;q¹⁼² u₁

iN

2 logq;q¹⁼² ˆ

eb;N;q g1 N;b‡N;Nq

N

QE

N²‡N 2

( 1)^N u₁ iNlogq;q¹⁼² u₁

iN

2 logq;q¹⁼² : …85†

On the otherhand, by (5), (83) and (84) we have

…86† hN;qi_N g1 N;q

N

QE

N²‡N 2

( 1)^N u1 iNlogq;q¹⁼² u1

iN

2 logq;q¹⁼² ˆD1

2;q²E

N:

The last two formulas together prove (78). p

4. Multiple extensions of Gauss' formula.

In this section we will look at several multipleq-formulas, so we start with some notation.

DEFINITION8. The notationP

~

m denotes a multiple summation with the indices m1;. . .;mn running over all non-negative integer values. In this connection we putjmj Pⁿ

jˆ1m_j.

Ifm~and~kare two arbitrary vectors withnelements, theirq-binomial coefficient is defined as

m~

~k _qYⁿ

jˆ1

m_j k_{j q}: …87†

Iffx_jgⁿ_jˆ1andfy_jgⁿ_jˆ1are two arbitrary sequences of complex numbers, then theirscalarproduct is defined by

~ xyXⁿ

jˆ1

x_jy_j: …88†

In the same way we define the following vectorversions of powers, etc.

~x^~aYⁿ

jˆ1

x^a_j^j; …89†

d_m;~~k 1; if m~ˆ~k;

0; otherwise:

( …90†

q ^~k2 Yⁿ

jˆ1

q ^kj2

; …91†

(~u;q)~s Yⁿ

jˆ1

(uj;q)_sj; ( 1)^~k ( 1)^jkj: …92†

The following theorem, fornˆ1 forms the basis ofq-analysis.

According to Jensen [26, p. 30], Ward [44, p. 255] and Kupershmidt [29, p. 244], this theorem was obtained by Euler. It was also obtained by Gauss and published posthumously in 1876 [19]. It is proved by induction [12].

THEOREM4.1.

X^~k

mˆ~ ~0

( 1)^~m ~k m~ !

q

q… †^m~2 ~u^m~ ˆ(~u;q)~k

…93†

COROLLARY4.2.

X^~k

~ mˆ~0

( 1)^m~ ~k

~ m !

q

q… † ˆ^m~2 d~k;~0

…94†

Ouraim is now to find some furthermultidimensional variations of (94) and a couple of related formulas. For the proof we need some other for- mulas, starting with

THEOREM4.3. The Jackson q-derivative expressed as q-difference of a q-shifted factorial.

Dⁿ_q;qahg‡a;qi_k ˆ( 1)ⁿfk n‡1g_n;qhg‡a‡n;qi_{k n}q… †^{n2 ‡ng}: …95†

We will only need oneq-Lauricella function, which is defined by DEFINITION9.

F⁽ⁿ⁾_D (a;b₁;. . .;b_n;cjq;x₁;. . .;x_n)ˆ

ˆX

~ m

ha;qi_m1‡...‡mnhb₁;qi_m1. . .hb_n;qi_mnQⁿ

jˆ1x^m_j ^j hc;qi_m1‡...‡mnQⁿ

jˆ1h1;qi_mj : …96†

The following lemma is aq-analogue of Chaundy [10, p. 164].

LEMMA4.4.

X^R

rˆ0

X^S

sˆ0

( 1)^r‡sh1;qi_Rh1;qi_S h1;qi_rh1;qi_{R r}h1;qi_sh1;qi_{S s} hc 1;qi_r‡shc;qi_2r‡2s

hc 1;qi_2r‡2shc;qi_R‡S‡r‡s QE r

2 ‡ s 2 ‡Sr

ˆ 1; if RˆSˆ0;

0; otherwise:

…97†

PROOF. LHSˆX^R

rˆ0

X^S

sˆ0

h R;qi_rh S;qi_s h1;qi_rh1;qi_sDc 1

2 ;c 2;q²E

r‡s

hc 1;qi_r‡sDc‡1 2 ;c

2;q²E

hc;qi_R‡Shc‡R‡S;qi_r‡sr‡sQE(Rr‡S(r‡s))

ˆ 1 hc;qi_R‡S

X

R‡S kˆ0

hc 1;qi_kDc‡1 2 ;c

2;q²E

k

hc‡R‡S;qi_kDc 1 2 ;c

2;q²i_k X

r‡sˆk

h R;qi_rh S;qi_s

h1;qi_rh1;qi_s QE(Rr‡Sk) …98†

ˆ 1 hc;qi_R‡S

X

R‡S kˆ0

hc 1;qi_kDc‡1 2 ;c

2;q²E

k

hc‡R‡S;qi_kDc 1 2 ;c

2;q²E

k

X^k

rˆ0

h R; k;qi_rh S;qi_k

h1;1‡S k;qi_rh1;qi_kQE(Rr‡Sk‡r(1‡S))

ˆ 1 hc;qi_R‡S

X

R‡S kˆ0

hc 1;qi_kDc‡1 2 ;c

2;q²E

k

hc‡R‡S;qi_kDc 1 2 ;c

2;q²E

k

h S;qi_k

h1;qi_k ²f₁( k; R;1‡S kjq;q^1‡S‡R)ˆ

ˆ 1

hc;qi_R‡S X

R‡S kˆ0

hc 1;qi_kDc‡1 2 ;c

2;q²E

k

hc‡R‡S;qi_kDc 1 2 ;c

2;q²E

k

h S;qi_k

h1;qi_k Gq 1‡S k;1‡S‡R 1‡S;1‡S k‡R

ˆ

ˆ 1

hc;qi_R‡S X

R‡S kˆ0

hc 1;qi_kDc‡1 2 ;c

2;q²E

k

hc‡R‡S;qi_kDc 1 2 ;c

2;q²E

k

(98)

h S;qi_k h1;qi_k

h1‡S;qi _k h1‡R‡S;qi _kˆ

ˆ 1

hc;qi_R‡S X

R‡S kˆ0

Dc 1; S R;c‡1 2 ;cg‡1

2 ;qE

k q^k(S‡R) hc‡R‡S;1;c 1

2 ;cg1 2 ;qE

k

ˆ

ˆ 1

hc;qi_R‡S ⁴f₃(c 1; R S;c‡1 2 ;cg‡1

2 ;c‡R‡S;c 1 2 ;cg1

2 jq;q^S‡R)ˆ

ˆ 1

hc;qi_R‡SGq

R‡S;c‡R‡S;c‡1 2 ;c 1

2 c‡1

2 ‡R‡S;c 1

2 ‡R‡S;0;c 2

66 4

3 77 5:

Result 0, except forR‡Sˆ0. Then the 3f₂equals 1 and we get 1. p COROLLARY4.5.

X^s

mˆ0

X^t

nˆ0

( 1)^m‡n s m q

t

n _q q… †^{m2 ‡}… †ⁿ2 ^‡mtˆ 1; if sˆtˆ0;

0; otherwise:

…99†

The following result from [13] leads to the same formula.

F1(a;b;b⁰;cjq;q^{c a b};q^{c a b b0})ˆ

F₁(a;b;b⁰;cjq;q^{c a b b0};q^{c a b0})ˆG_q c;c a b b⁰ c a;c b b⁰

…100† ;

wherejq^{c a b b0}j<1. Putbˆ s;b⁰ˆ tto obtain X^s

mˆ0

X^t

nˆ0

( 1)^m‡n s m q

t

n _qha;qi_m‡nhc‡m‡n;qi_{s‡t m n}

QE m

2 ‡ n

2 ‡m(c a)‡n(c a‡s)

ˆ hc a;qi_s‡t: …101†

Now multiply withq^a(s‡t)and applyD^s‡t_q;qa to both sides to obtain X^s

mˆ0

X^t

nˆ0

s m q

t

n _qhc‡m‡n;qi_{s‡t m n} QE m‡n

2

m

2 ‡ n

2 ‡c(m‡n)‡ns

ˆ1 …102†

Finally, applyD^s‡t_q;qcto both sides to obtain X^s

mˆ0

X^t

nˆ0

( 1)^m‡n s m q

t

n _qQE m‡n 2

‡ m

2 ‡ n

2 ‡ns

QE s‡t m n 2

‡(s‡t m n)(m‡n)

ˆ0 …103†

The following important formula is aq-analogue of [34] and [37].

THEOREM 4.6. [14]. If fC_ng¹_nˆ0; fa_ng¹_nˆ0 are bounded sequences of complex numbers then

X

~ m

C_m1‡...‡mnQⁿ

jˆ1x^mjha_j;qi_mj Qⁿ

jˆ1h1;qi_mj QE Xⁿ

kˆ1

m_kX^k

lˆ2

a_l

!

ˆ

X¹

Nˆ0

CNx^ND Pⁿ

kˆ1ak;qE

N

h1;qi_N QE NXⁿ

lˆ2

al

! : …104†