A note on Bailey and WP-Bailey Pairs

Satya Prakash Singh 1 , Lakshmi Narayan Mishra 2 ,  and Vijay Yadav 3
  • 1 Department of Mathematics, TDPG College, , Jaunpur, India
  • 2 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore, India
  • 3 Department of Mathematics, SPDT College, , Andheri(E), Mumbai, India
Satya Prakash Singh, Lakshmi Narayan Mishra
  • Corresponding author
  • Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore, 632 014, Tamil Nadu, India
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and Vijay Yadav

Abstract

In this paper, we have established certain theorems involving Bailey pairs and WP-Bailey pairs. Further, making use of some known WP-Bailey pairs and theorems for constructing new WP-Bailey pairs, we have also established transformation formulas for q-hypergeometric series.

1 Introduction, Notations and Definitions

We begin by recalling some standard notations and terminology. Let a, q be complex numbers with 0 < |q| < 1. Then the q − shifted factorial is defined by

(a;q)n=(1a)(1aq)...(1aqn1)   if  n>0,(a;q)0=1
and
(a;q)=r=0(1aqr).

For the sake of brevity, we often write

(a1;q)n(a2;q)n...(ar;q)n=(a1,a2,a3,...,ar;q)n.

The basic hypergeometric series is defined by

rΦs=[a1,a2,...,ar;q;zb1,b2,...,bs]=n=0(a1,a2,...,ar;q)nzn(q,b1,b2,...,bs;q)n{(1)nqn(n1)/2}1+sr.

In an attempt to clarify Rogers second proof [3] of the Rogers-Ramanujan identities, Bailey [1] made the following simple but very useful observation,

If

βn=r=0nαrunrvn+r
and
γn=r=nδrurnvr+n=r=0δr+nurvr+2n
then
n=0αnγn=n=0βnδn.

The proof is straightforword and merely requires an interchange of sums. Of course, in the above transform, suitable convergence conditions need to be imposed to make the definition of γn and interchange of sums meaningful.

In application of the transform, Bailey chose ur=1(q;q)r , vr=1(aq;q)r and with this choice equations (1.1) and (1.2) became

βn=r=0nαr(q;q)nr(aq;q)n+r
and
γn=r=0δr+n(q;q)r(aq;q)r+2n
respectively.

A pair of sequence 〈αn, βn〉 that satisfies (1.4) is called a Bailey pair relative to the parameter a. Similarly, the pair of sequence 〈γn, δn〉 which satisfies (1.5) is called conjugate Bailey pair relative to a. For these Bailey and conjugate Bailey pairs we have

n=0αnγn=n=0βnδn,
provided series involved are convergent.

2 Theorems Involving Bailey Pairs

(i) Choosing δr=(ρ1,ρ2;q)r(aqρ1ρ2)r in (1.5) and using the summation formula [2, App. II(II.8) p. 236] we have,

γn=(aqρ1,aqρ2;q)(ρ1,ρ2;q)n(aqρ1ρ2)n(aq,aqρ1ρ2;q)(aqρ1,aqρ2;q)n.

Putting these values of γn and δn in (1.6) we obtain the following theorem.

Theorem 1

If 〈αn, βnis a Bailey pair satisfying (1.4) then

n=0(ρ1,ρ2;q)n(aqρ1ρ2)n(aqρ1,aqρ2;q)nαn=(aq,aqρ1ρ2;q)(aqρ1,aqρ2;q)n=0(ρ1,ρ2;q)n(aqρ1ρ2)nβn.

Taking ρ1, ρ2 → ∞ in (2.2) we obtain the following theorem,

Theorem 2

If 〈αn, βn〉 is a Bailey pair then

n=0qn2anαn=(aq;q)n=0qn2anβn.

(ii) Choosing δr=(ρ1,ρ2;q)r(aρ1ρ2)r in (1.5) and using the summation formula [4, (1.4) p. 771] we have,

γn=(aqρ1,aqρ2;q)(aq,aqρ1ρ2;q){(ρ1,ρ2;q)n(aρ1ρ2)n(aqρ1,aqρ2;q)n+(1ρ1)(1ρ2)(ρ1ρ2a)(ρ1q,ρ2q;q)n(aρ1ρ2)n(aqρ1,aqρ2;q)n}.

Putting these values of γn and δn in (1.6) we have following theorem.

Theorem 3

If 〈αn, βn〉 is a Bailey pair then

n=0(ρ1,ρ2;q)n(aρ1ρ2)n(aqρ1,aqρ2;q)nαn+(1ρ1)(1ρ2)a(ρ1ρ2a)n=0(ρ1q,ρ2q;q)n(aρ1ρ2)n(aqρ1,aqρ2;q)nαn=(aq,aqρ1ρ2;q)(aqρ1,aqρ2;q)nn=0(ρ1,ρ2;q)n(aρ1ρ2)nβn.

Taking ρ1, ρ2 → ∞ in (2.5) we have following theorem,

Theorem 4

If 〈αn, βn〉 is a Bailey pair then

n=0qn(n1)anαn+an=0qn(n+1)anαn=(aq;q)n=0qn(n1)anβn

(iii) Taking δr=(ρ1,ρ2;q)r(aqρ1ρ2;q)Nr(q;q)Nr(aqρ1ρ2)r in (1.5) and using the summation formula [2, App. II (II. 12), p. 237] we obtain

γn=(aqρ1,aqρ2;q)N(ρ1,ρ2;q)n(qN;q)n(aq1+Nρ1ρ2)n(q,aq;q)Nqn(n1)/2(aqρ1,aqρ2,aq1+N;q)n.

Putting these values of γn and δn in (1.6) we have,

Theorem 5

If 〈αn, βn〉 is a Bailey pair then

n=0N(ρ1,ρ2,qN;q)n(aq1+Nρ1ρ2)nαnqn(n1)/2(aqρ1,aqρ2,aq1+N;q)n=(aq,aqρ1ρ2;q)N(aqρ1,aqρ2;q)Nn=0N(ρ1,ρ2,qN;q)nqn(ρ1ρ2aqN;q)nβn.

For N → ∞, (2.8) yields (2.2).

3 WP-Bailey Pairs and Related Theorems

In this section we have established certain theorems involving WP-Bailey pairs.

A WP-Bailey is a pair of sequences {αn(a,k), βn(a,k)} satisfying α0(a,k) = β0(a,k) = 1 and

βn(a,k)=r=0n(ka;q)nr(k;q)n+r(q;q)nr(aq;q)n+rαr(a,k)=(ka,k;q)n(q,aq;q)nr=0n(qn,kqn;q)r(aq1nk,aq1+n;q)r(aqk)rαr(a,k).

The corresponding WP-conjugate Bailey pair 〈γn(a,k), δn(a,k)〉 is given by,

γn(a,k)=r=0(ka;q)r(k;q)r+2n(q;q)r(aq;q)r+2nδr+n(a,k)=(k;q)2n(aq;q)2nr=0(ka;q)r(kq2n;q)r(q;q)r(aq1+2n;q)rδr+n(a,k).

If 〈αn(a,k), βn(a,k)〉 is a WP-Bailey pair and 〈γn(a,k), δn(a,k)〉 is a WP-Conjugate Bailey pair then Bailey lemma gives,

n=0αn(a,k)γn(a,k)=n=0βn(a,k)δn(a,k),
provided series involved in (3.2) and (3.3) are convergent.

(i) Choosing δr(a,k)=(a2qk2)r in (3.2) and using the summation formula [2, App. II(II.8), p. 236] we find,

γn(a,k)=(a2qk,aqk;q)(aq,a2qk2;q)(k;q)2n(a2qk;q)2n(a2qk2)n.

Putting these values of 〈γn(a,k), δn(a,k)〉 in (3.3) we have following theorem.

Theorem 6

If 〈αn(a,k), βn(a,k)〉 is a WP-Bailey pair then

(a2qk,aqk;q)(aq,a2qk2;q)n=0(k;q)2n(a2qk;q)2n(a2qk2)nαn(a,k)=n=0(a2qk2)nβn(a,k).

(ii) Putting δr(a,k)=(a2k2)r in (3.2) and making use of the summation formula [4, (1.4), p.771] we get,

γn(a,k)=(a2qk,aqk;q)(aq,a2qk2;q)(k;q)2n(a2qk;q)2n(kk+a)(1+aq2n)(a2k2)n.

Substituting these values 〈γn(a,k), δn(a,k)〉 in (3.3) we get the following theorem.

Theorem 7

If 〈αn(a,k), βn(a,k)〉 is a WP-Bailey pair then

(a2qk,aqk;q)(aq,a2qk2;q)(kk+a)n=0(k;q)2n(a2qk;q)2nαn(a,k)(a2k2)n+(a2qk,aqk;q)(aq,a2qk2;q)(kak+a)n=0(k;q)2n(a2qk;q)2n(a2q2k2)nαn(a,k)=n=0βn(a,k)(a2k2)n.

(iii) Taking δr=(a2qk2;q)Nr(a2qk2)r(q;q)Nr in (3.2) we find,

γn(a,k)=(k;q)2n(aq;q)2n(a2qk2;q)Nn(q;q)Nn(a2qk2)n 3Φ2[kq2n,ka,q(Nn);q;qaq1+2n,k2a2q(Nn)].

Now summing 3Φ2 series in (3.8) by making use of [2, App. II(II.12), p. 237] we get,

γn(a,k)=(a2qk,aqk;q)N(q,aq;q)N(aq;q)2n(a2qk;q)2n(qN,a2q1+Nk;q)n(aqk)n(aq1+N,kaqN;q)n.

Putting these values of 〈γn(a,k), δn(a,k)〉 in (3.3) we have following theorem.

Theorem 8

Ifαn(a,k), βn(a,k)〉 is a WP-Bailey pair then

n=0N(k;q)2n(qN,a2kq1+N;q)n(aqk)n(a2qk;q)2n(kqNa,aq1+N;q)nαn(a,k)=(aq,a2qk2;q)N(a2qk,aqk;q)Nn=0N(qN;q)nqn(k2qNa2;q)nβn(a,k).

(iv) Choosing δr(a,k)=(1kq2r)(1k)(ρ1,ρ2;q)r(aqρ1ρ2)r(kqρ1,kqρ2;q)r in (3.2) we find,

γn(a,k)=(k;q)2n(ρ1,ρ2;q)n(1kq2n)(aqρ1ρ2)n(aq;q)2n(kqρ1,kqρ2;q)n(1k). 6Φ5[kq2n,qn+1k,qn+1k,ρ1qn,ρ2qn,ka;q;aqρ1ρ2qnk,qnk,kqn+1ρ1,kqn+1ρ2,aq1+2n].

Summing the 6Φ5 series in (3.11) by using [2, App. II (II 20), p. 238] we find,

γn(a,k)=(kq,kqρ1ρ2,aqρ1,aqρ2;q)(ρ1,ρ2;q)n(kqρ1,kqρ2,aq,aqρ1ρ2;q)(aqρ1,aqρ2;q)n(aqρ1ρ2)n.

Putting these values of 〈γn(a,k), δn(a,k)〉 in (3.3) we get following theorem.

Theorem 9

Ifαn(a,k), βn(a,k)〉 is a WP-Bailey pair then

(kq,kqρ1ρ2,aqρ1,aqρ2;q)(kqρ1,kqρ2,aq,aqρ1ρ2;q)n=0(ρ1,ρ2;q)n(aqρ1ρ2)n(aqρ1,aqρ2;q)nαn(a,k)=n=0(1kq2n1k)(ρ1,ρ2;q)n(aqρ1ρ2)n(kqρ1,kqρ2;q)nβn(a,k).

As ρ1, ρ2 → ∞, (3.13) yields following theorem.

Theorem 10

Ifαn(a,k), βn(a,k)〉 is a WP-Bailey pair then

(kq;q)(aq;q)n=0qn2anαn(a,k)=n=0(1kq2n1k)qn2anβn(a,k).

(v) Taking δr(a,k)=(ρ1,ρ2,akρ1ρ2q1+N;q)r(1k;q)Nr(kqρ1,kqρ2,ρ1ρ2aqN;q)r(q;q)Nr(q2Nk)r in (3.2) we get

γn(a,k)=(k;q)2n(1kq2n)(ρ1,ρ2,akρ1ρ2q1+N;q)n(qN;q)nqn(aq;q)2n(1k)(kqρ1,kqρ2,ρ1ρ2aqN;q)n(kq1+N;q)nqN(N+1)/2kN(q,kq;q)N.8Φ7[kq2n,qn+1k,qn+1k,ρ1qn,ρ2qn,akρ1ρ2q1+N+n,ka,q(Nn);q;qqnk,qnk,kρ1q1+n,kρ2q1+n,ρ1ρ2aq(Nn),aq1+2n,kq1+N+n].

Summing the 8Φ7 series in (3.15) by using [2, App.II (II.22), p. 238] we have

γn(a,k)=(kqρ1ρ2,aqρ1,aqρ2;q)NqN(N+1)/2kN(q,aqρ1ρ2,kqρ1,kqρ2;q)N(aq;q)N .(ρ1,ρ2,kqρ1,kqρ2,akρ1ρ2q1+N,qN;q)n(aqk)n(aq1+N,aqρ1,aqρ2,ρ1ρ2kqN,kqρ1,kqρ2;q)n.

Putting these values of 〈γn(a,k), δn(a,k)〉 in (3.3) we obtain the following theorem.

Theorem 11

Ifαn(a,k), βn(a,k)〉 is a WP-Bailey pair then

(kq,kqρ1ρ2,aqρ1,aqρ2;q)N(aq,aqρ1ρ2,kqρ1,kqρ2;q)Nn=0N(ρ1,ρ2,akρ1ρ2q1+N,qN;q)n(aqk)n(aq1+N,aqρ1,aqρ2,ρ1ρ2kqN;q)n=n=0(1kq2n1k)(ρ1,ρ2,akρ1ρ2q1+N,qN;q)nqn(kq1+N,kqρ1,kqρ2,ρ1ρ2aqN;q)nβn(a,k).

As ρ2 → ∞, (3.17) yields the following theorem.

Theorem 12

Ifαn(a,k), βn(a,k)〉 is a WP-Bailey pair then

(kq,aqρ1;q)N(aq,kqρ1;q)Nn=0N(ρ1,qN;q)n(aq1+Nρ1)nαn(a,k)(aqρ1,aq1+N;q)n=n=0N(1kq2n1k)(ρ1,qN;q)n(kqρ1,kq1+N;q)n(aρ1q1+N)nβn(a,k).

4 Bailey Pairs

In this section we give numerous Bailey pairs deducible from certain summation formulas.

(i) Choosing αr=qr(r+1)/2(a,qa,qa;q)rq32r(q,a,a;q)r in (1.4) and using the summation formula [5, (4.1), p.76] we get,

βn=12[(q12;q)n(1+a)(q,aq,a;q)n+(q12;q)n(1a)(q,aq,a;q)n].
αn and βn given in (4.1) form a Bailey pair.

(ii) Taking αr=qr(r+1)/2(a,qa;q)rqr(q,a;q)r in (1.4) and using the summation formula [5, (4.2), p.76] we get,

βn=1+a2(1;q)n(q,aq,aq;q)n+1a2(1;q)n(q,a,qa;q)n.
αn and βn given in (4.2) form a Bailey pair.

(iii) Taking αr=qr(r+1)/2(a;q)rq12r(q;q)r in (1.4) and making use of the summation formula [5, (4.3), p.76] we find,

βn=1+a2(q12;q)n(q,aq,qa;q)n+1a2(q12;q)n(q,aq,qa;q)n.
αn and βn given in (4.3) form a Bailey pair.

(iv) Choosing αr=q12r2(a;q)r(1aq2r)(q;q)r(1a) in (1.4) and using the summation formula [5, (4.5), p.77] we find,

βn=1+a2a(q12;q)n(q,aq,qa;q)n1a2a(q12;q)n(q,aq,qa;q)n.
αn and βn given in (4.4) form a Bailey pair.

(v) Taking αr=qr(r+1)/2(a,qa;q)r(q,a;q)r in (1.4) and using the summation formula [5, (4.6), p.77] we find,

βn=1+a2a(1;q)n(q,aq,aq;q)n1a2a(1;q)n(q,a,qa;q)n.
αn and βn given in (4.5) form a Bailey pair.

(vi) Choosing αr=qr(r+1)/2(a;q)rq12r(q;q)r in (1.4) and making use of the summation formula [5, (4.7), p.77] we find,

βn=1+a2a(q12;q)n(q,aq,qa;q)n1a2a(q12;q)n(q,aq,qa;q)n.
αn and βn given in (4.6) form a Bailey pair.

(vii) Taking αr=qr(r+1)/2(a,qa,b;q)r(ab)r(q,a,aqb;q)r in (1.4) and using the summation formula [2, App. II (II.14), p. 237] we get

βn=(qab;q)n(q,qa,aqb;q)n.
αn, βn〉 given in (4.7) form a Bailey pair.

(viii) Taking αr=qr(r+1)/2(a,b,c;q)r(1aq2r)(abc)r(q,aqb,aqc;q)r(1a) in (1.4) and using the summation formula [2, App. II (II.21), p. 238] we get

βn=(aqbc;q)n(q,aqb,aqc;q)n.
αn, βn〉 given in (4.8) form a Bailey pair.

5 WP-Bailey Pairs

In this section we give certain WP-Bailey pairs out of which some are known and some are new.

(i) Taking αr(a,k)=(a,b;q)r(1aq2r)(q,aqb;q)r(1a)(1b)r in (3.1) and summing the series by making use of [2, App. II (II.21), p. 238] we get,

βn(a,k)=(k,kba;q)n(q,aqb;q)nbn.
αn(a,k), βn(a,k)〉 given in (5.1) form a WP-Bailey pair.

(ii) Choosing αr(a,k)=(a,b,c,a2qbck;q)r(1aq2r)(q,aqb,aqc,bcka;q)r(1a)(ka)r in (3.1) and using the summation formula [2, App. II (II.22), p. 238] we get,

βn(a,k)=(k,aqbc,kba,kca;q)n(q,aqb,aqc,kbca;q)n.
αn(a,k), βn(a,k)〉 given in (5.2) form a WP-Bailey pair.

(iii) Again, choosing αn(a,k)=(a,ak;q)n(1aq2n)(q,kq;q)n(1a)(ka)n in (3.1) and summing the series by making use of [2, App. II (II.21), p. 238] we get,

βn(a,k)={1,     n=0.0,     n1

So, 〈αn(a,k), βn(a,k)〉 given in (5.3) form a WP-Bailey pair.

(iv) If we take αr(a,k) = δr,0 in (3.1) we find,

βn(a,k)=(k,ka;q)n(q,aq;q)n.
αn(a,k), βn(a,k)〉 given in (5.4) also form a WP-Bailey pair.

(v) In the summation formula [5, (1.3), p.71] if we take c=akq1/2 we get,

4Φ3[a,akq12,kqn,qn;q;qkq12,akq1n,aq1+n]=1+a2(aq,q;q)n(ka,kqa;q)n(kq12,ka;q)n(aq,qa;q)n+1a2(aq,q;q)n(ka,kqa;q)n(kq12,ka;q)n(aq,qa;q)n.

Now, choosing αr(a,k)=(a,aq12k;q)n(q,kq12;q)n(ka)n in (3.1) and summing the series by using (5.5) we get

βn(a,k)=1+a2(k,q12,ka,kqa;q)n(q,kq12,qa,aq;q)n+1a2(k,q12,ka,kqa;q)n(q,kq12,qa,aq;q)n.
αn(a,k), βn(a,k)〉 given in (5.6) form a WP-Bailey pair.

(vi) Again, taking c=akq12 in the summation formula [5, (4.4), p.77] we get,

4Φ3[a,akq12,kqn,qn;q;q2kq12,akq1n,aq1+n]=1+a2a(aq,q;q)n(ka,kqa;q)n(kq12,kq12a;q)n(aq,qa;q)n+1a2a(aq,a;q)n(ka,kqa;q)n(kq12,kq12a;q)n(aq,qa;q)n.

Now, choosing αn(a,k)=(a,aq12k;q)n(q,kq12;q)n(kqa)n in (3.1) and using (5.7) we get

βn(a,k)=1+a2a(k,q12,ka,kqa;q)n(q,kq12,qa,aq;q)n1a2a(k,q12,ka,kqa;q)n(q,kq12,qa,aq;q)n.
αn(a,k), βn(a,k)〉 given in (5.8) form a WP-Bailey pair. Bailey pairs given in (5.6) and (5.8) are believed to be new.

6 Bailey Chain

If 〈αn, βn〉 is a Bailey pair i.e. βn=r=0nαr(q;q)r(aq;q)r , then so 〈 αn' , βn' 〉 is also a Bailey pair, where

αn=(ρ1,ρ2;q)n(aqρ1ρ2)n(aqρ1,aqρ2;q)nαn.
and
βn=r=0(ρ1,ρ2;q)r(aqρ1ρ2;q)nr(aqρ1ρ2)rβr(q;q)nr(aqρ1,aqρ2;q)n=(aqρ1ρ2;q)n(q,aqρ1,aqρ2;q)nr=0(ρ1,ρ2;q)r(qn;q)rqr(ρ1ρ2aqn;q)rβr.

Thus, we find that if one Bailey pair 〈αn, βn〉 is known then a new Bailey pair 〈 αn' , βn' 〉 can be constructed as shown above in (6.2). Repeating this process we can have infinite number of Bailey pairs if one initial pair is known. These Bailey pairs so constructed from an initial Bailey pair form a chain called Bailey chain.

7 WP-Bailey Tree

Andrews proved following two theorems for constructing WP-Bailey pairs from a initial known Bailey pair.

Theorem 13

Ifαn(a,k), βn(a,k)〉 is a WP-Bailey pair, then so is the pairαn(a,k) , βn(a,k)given by,

αn(a,k)=(b,c;q)n(aqb,aqc;q)n(km)nαn(a,m),βn(a,k)=(mqb,mqc;q)n(aqb,aqc;q)nr=0n1mq2r1m(b,c;q)r(km;q)nr(k;q)n+r(mqb,mqc;q)r(q;q)nr(mq;q)m+r(km)rβr(a,m),

[6, Theorem (2.1)]

where m=bckaq .

Theorem 14

Ifαn(a,k), βn(a,k)〉 is a WP-Bailey pair, then so is the pairαn(a,k) , βn(a,k)given by,

αn(a,k)=(m;q)2n(k;q)2m(km)nαn(a,m),βn(a,k)=r=0n(km;q)nr(q;q)nr(km)rβr(a,m),

[6, Theorem (2.2)]

where m=a2qk .

From these two theorems, each WP-Bailey pair gives rise to a binary tree of WP-Bailey pairs. Andrews coined this the WP-Bailey tree. The following four theorems due to Warnaar give additional branches to the Bailey tree.

Theorem 15

Ifαn(a,k), βn(a,k)〉 is a WP-Bailey pair, then so is the pairαn(a,k) , βn(a,k)given by,

αn(a,k)=1σk121σk12qn1+σm12qn1+σm12(m;q)2n(k;q)2m(km)nαn(a,m),βn(a,k)=1σk121σk12qnr=0n1+σm12qn1+σm12(km;q)nr(q;q)nr(km)rβr(a,m),

[6, Theorem (2.3)]

where m=a2k and σ = {−1, 1}.

The freedom in the choice of σ simply reflects that the above expressions are invariant under the simultaneous negation of k12 , m12 and σ.

Theorem 16

Ifαn(a,k), βn(a,k)〉 is a WP-Bailey pair, then so is the pairαn(a,k) , βn(a,k)given by,

αn(a2,k;q2)=αn(a,m;q),βn(a2,k;q2)=(mq;q)2n(aq;q)2nr=0n(1mq2r)(1m)(km2;q2)nr(k;q2)n+r(ma)nr(q2;q2)nr(m2q2;q2)n+rβr(a,m;q),

[6, Theorem (2.4)]

where m=kaq .

Theorem 17

Ifαn(a,k;q), βn(a,k;q)〉 is a WP-Bailey pair, then so is the pairαn(a,k;q) , βn(a,k;q)given by,

αn(a2,k;q2)=qn1+aq2n1+aαn(a,m;q),βn(a2,k;q2)=qn(mq;q)2n(a;q)2nr=0n(1mq2r)(1m)(km2;q2)nr(k;q2)n+r(ma)nr(q2;q2)nr(m2q2;q2)n+rβr(a,m;q),

[6, Theorem (2.5)]

where m=ka .

Theorem 18

Ifαn(a,k;q), βn(a,k;q)〉 is a WP-Bailey pair, then so is the pairαn(a,k;q) , βn(a,k;q)given by,

α2n(a,k;q)=αn(a,m;q2),   α2n+1(a,k;q)=0,βn(a,k;q)=(mq;q2)n(aq;q)nr=0[n2](1mq2r)(1m)(km;q)nr(k;q)n+2r(ka)n2r(q;q)nr(mq;q)n+2rβr(a,m;q2),

[6, Theorem (2.6)]

where m=ka .

8 Applications

In this section we shall establish certain transformation formulas by making use of the results established in previous sections.

(a) Substituting the Bailey pairs given in (4.1) in (2.2) we get,

(aqρ1,aqρ2;q)(aq,aqρ1ρ2;q)5Φ5[a,qa,qa,ρ1,ρ2;q;aq12ρ1ρ2a,a,aqρ1,aqρ2,0]=1+a2 3Φ2[ρ1,ρ2,q12;q;aqρ1ρ2aq,a]+1a2 3Φ2[ρ1,ρ2,q12;q;aqρ1ρ2aq,a],   |aq12ρ1ρ2|<1.

Similar transformations can be established by putting Bailey pairs given in (4.1), (4.2), (4.3), (4.4), (4.5), (4.6), (4.7) and (4.8) in any one of the results given in (2.2), (2.3), (2.5), (2.6) and (2.8).

(b) If we put WP-Bailey pair given in (5.1) and (3.5) we get,

(a2qk,aqk;q)(aq,a2qk2;q)8Φ7[a,qa,qa,b,k,k,kq,kq;q;a2qbk2a,a,aqb,aqk,aqk,aqk,aqk]=2Φ1[k,kba;q;a2qbk2aqb].

Similar transformations can be established by substituting WP-Bailey pairs given in (5.2), (5.3), (5.4), (5.6) and (5.8) in any one of the results given in (3.5), (3.7), (3.10), (3.13), (3.14), (3.17) and (3.18).

(c) Replacing b, c by ρ1 and ρ2 in (7.1) respectively and then putting the values of αn(a, m), βn(a, m) from (5.1) we get new Bailey pairs.

αn(a,k)=(a,qa,qa,b,ρ1,ρ2;q)n(q,a,a,aqb,aqρ1,aqρ2;q)n(kmb)n,βn(a,k)=(k,km,mqρ1,mqρ2;q)n(q,mq,aqρ1,aqρ2;q)nr=0n1mq2r1m(m,mba,ρ1,ρ2,kqn,qn;q)r(qb)r(q,aqb,mqρ1,mqρ2,mkq1n,mq1+n;q)r,
where m=kρ1ρ2aq .

Now, putting these values of αn(a,k) and βn(a,k) given in (8.3) in (3.1) we get,

(km,mqρ1,mqρ2,aq;q)n(mq,aqρ1,aqρ2,ka;q)n8Φ7[m,qm,qm,mba,ρ1,ρ2,kqn,qn;q;qbm,m,aqb,mqρ1,mqρ2,mkq1n,mq1+n]=8Φ7[a,qa,qa,b,ρ1,ρ2,kqn,qn;q;aqmba,a,aqb,aqρ1,aqρ2,akq1n,aq1+n].
where m=kρ1ρ2aq .

Putting the WP-Bailey pairs given in (5.1), (5.2), (5.3), (5.4), (5.5), (5.6), (5.7) and (5.8) in any one of the results given in (7.1), (7.2), (7.3), (7.4), (7.5) and (7.6) one finds new WP-Bailey pairs, on substituting these new Bailey pairs in (3.1) we get transformations similar to (8.4).

9 Conclusions

In this paper, certain transformation formulas involving q-hypergeometric series have been obtained by making use of theorems, Bailey Pairs and WP-Bailey Pairs established herein. From these transformation formulas q-series identities can be deduced which may have partition theoretic interpretations. Results of this paper are quite useful and we hope that these results will form the base of further research in the subject.

References

  • [1]

    Bailey, W.N., Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 50 (1949), p. 1–10.

  • [2]

    Gasper, G. and Rahman, M., Basic hypergeometric series, Cambridge University Press, New York (1991).

  • [3]

    Rogers, L.J., On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc. (2) 16 (1917), p. 315–336.

  • [4]

    Verma, A., On identities of Rogers-Ramanujan type, Indian J. Pure Appl. Math., 11(6), (1980) p. 770–790.

  • [5]

    Verma, A. and Jain, V.K., Certain summation formulae for q-series, J. Indian Math. Soc., 47(1983), p. 71–85.

  • [6]

    Warnaar, S. OLE, Extensions of the well poised and elleptic well-poised Bailey lemma, Indge Math. (N.S.) 14 (2003), p. 571–588.

    • Crossref
    • Export Citation

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  • [1]

    Bailey, W.N., Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 50 (1949), p. 1–10.

  • [2]

    Gasper, G. and Rahman, M., Basic hypergeometric series, Cambridge University Press, New York (1991).

  • [3]

    Rogers, L.J., On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc. (2) 16 (1917), p. 315–336.

  • [4]

    Verma, A., On identities of Rogers-Ramanujan type, Indian J. Pure Appl. Math., 11(6), (1980) p. 770–790.

  • [5]

    Verma, A. and Jain, V.K., Certain summation formulae for q-series, J. Indian Math. Soc., 47(1983), p. 71–85.

  • [6]

    Warnaar, S. OLE, Extensions of the well poised and elleptic well-poised Bailey lemma, Indge Math. (N.S.) 14 (2003), p. 571–588.

    • Crossref
    • Export Citation
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