A reduction formula for a q beta integral

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R E S E A R C H

Open Access

A reduction formula for a

q

-beta integral

Zeinab Sayed Mansour

1,2

_{and Maryam Ahmed Al-Towailb}

3*

*_{Correspondence:}

mtowaileb@ksu.edu.sa

3_{Department of Natural and}

Engineering Science, Faculty of Applied Studies and Community Service, King Saud University, Riyadh, KSA

Full list of author information is available at the end of the article

Abstract

In this paper, we give a reduction formula for a speciﬁcq-integral. Our formula is expressed as a three term recurrence relations for basic hypergeometric3

φ

2series. This is aq-analog of work by Watson and by Bailey of 1953.

Keywords: q-beta integral; basic hypergeometric series; contiguous relations

1 Introduction and preliminaries

In [], Watson constructed a reduction formula for the integral

In=

 

xα_{( –}_x₎βdn{x

γ_{( –}_x₎δ_}

dxn dx.

In fact, he proved that

In=

(α+γ –n+ )(β+δ–n+ )

(σ–n+ ) Hn,

where

Hn= n

r=

(–)n–r

n r

(–γ)n–r(β+δ–n+ )n–r(–δ)r(α+γ –n+ )r,

and the notation (x)rdenotes the product

(x)≡ and (x)r≡x(x+ )· · ·(x+r– ), r≥.

Furthermore, Watson proved thatHnsatisﬁes the following three term recurrence

rela-tion:

(σ–n)(σ–n)Hn+–QnHn++SnHn= , (.)

where

Sn= (n+ )(σ–n– )(α+β–n)(γ+δ–n)(β+δ–n),

Qn= (σ– n– )

(βγ–αδ)(σ+ ) +θ(n+ )(σ–n)

,

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and σ=α+β+γ+δ, θ=α–β–γ+δ. Also, he remarkedHncan be expressed in terms

of a hypergeometric series of the typeFwith last element unity which implies that (.)

gives a three term contiguous relation for terminatingFseries.

The proof introduced by Watson depends on constructing a second order linear diﬀer-ential equation satisﬁed by the integrand ofIn. On the other hand, in [], Bailey derived

relations between contiguous hypergeometric functions of the typeF(), and by using

these relations, he obtained another proof of Watson’s reduction formula. In this paper, we introduce aq-analog of the integralInby

In,q=

 

xα(qx;q)βDn_q–

xγ qβ+x;q_δdqx, n= , , , . . . ,

whereα,β,γ, andδare complex numbers, andqis a positive number less than one. Our aim to obtain a reduction formula forIn,q.

It turns out to us that Watson technique for introducing (.) is too hard to applied to our work. Therefore, we follow Bailey’s approach for deriving the reduction formula.

We recall the following deﬁnitions (see,e.g., [–]): Theq-shifted fractional is deﬁned by

(a;q)∞= ∞

j=

 –aqj, and (a;q)n:=

(a;q)∞

(aqn_;_q₎_∞ forn∈Z,a∈C.

Theq-derivativeDqf of an arbitrary functionf is given by

(Dqf)(x) :=

f(x) –f(qx)

( –q)x , x= .

We follow Gasper and Rahman [] for the deﬁnitions of Jacksonq-integrals, and the

q-gamma andq-beta functions (see also [–]). Theq-integration by parts rule (see []) is

a



f(qt)Dqg(t)dqt=f(a)g(a) – lim n→∞f q

n_g _qn_–

a



Dqf(t)g(t)dqt.

Leta, . . . ,ar,b, . . . ,bsbe complex numbers, theq-hypergeometric seriesrφsdeﬁned by

rφs(a, . . . ,ar,b, . . . ,bs;q,z) = ∞

n=

(a, . . . ,ar;q)n

(q,b, . . . ,bs;q)n

zn –q(n–)/n(s–r+)

The series representation of the functionrφsconverges absolutely for allz∈Cifr≤s, and

converges only for|z|<  ifr=s+  (for more details and results see [–] and []). Observe that

I,q=

 

xα+γ(qx;q)β+δdqx=Bq(α+γ + ,β+δ+ )

=q(α+γ + )q(β+δ+ )

q(α+γ +β+δ+ )

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and, by using theq-integration by parts, one can verify that

I,q=

 

xα(qx;q)βDq–

xγ qβ+x;q_δdqx

=q–γ q(α+γ)q(β+δ)

q(α+γ +β+δ+ )

[γ][β+δ] –qβ_[_δ_][_α₊_γ_]_,

where the notation [z] is deﬁned by

[z] := –q

z

 –q.

Note that the above values of the integralsI,qandI,qcoincide withIandI, respectively,

which are given by Watson in the limitq→.

This paper is organized as follows. In Section , we derive three term contiguous rela-tions for the basic hypergeometric functionφ(a,b,c;d,e;q,q). In Section , we show that

In,qcan be represented asφ(q) and a direct substation in the derived contiguous relation

yields the result of this paper.

2 Contiguous relations of3

φ

2

Throughout this section, we simply usedato denotes the valueq–n_where_n_{is an arbitrary}

nonnegative integer. We denote byφthe function

φ(a,b,c;d,e;q,q),

and by φ(a+_), _φ₍_a–_{) the same function when} _a_{is changed to}_aq_,_a_/_q_{, respectively. We}

use a similar notation when the other parameters are so changed. Also, letφ+,φ–be the

functions deﬁned by

φ+=φ(aq,bq,cq;dq,eq;q,q) and φ–=φ(a/q,b/q,c/q;d/q,e/q;q,q).

By the deﬁnition ofφ, one can verify the following:

φ a+–φ=qa( –b)( –c)

( –d)( –e)φ+, (.)

φ a––φ= –a( –b)( –c)

( –d)( –e)φ+ a

–_, _(.)

φ d+–φ= –qd ( –a)( –b)( –c)

( –d)( –qd)( –e)φ+ d

+_, _(.)

φ d––φ=d ( –a)( –b)( –c)

( –d/q)( –d)( –e)φ+. (.)

These equations, and the symmetries of theφ, give us

c( –a)φ a+–φ=a( –c)φ c+–φ, (.)

a( –b)φ b+–φ=b( –a)φ a+–φ, (.)

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d( –e/q)φ e––φ=e( –d/q)φ d––φ. (.) Now, applying the transformation (see [])

φ(a,b,c;d,e;q,q) =

(de/bc;q)n

(e;q)n

(bc/d)nφ(a,d/b,d/c;d,de/bc;q,q), (.)

toψ=φ(a,d/b,d/c;qd,de/bc;q,q) yields the following relations:

φ=(de/bc;q)n (e;q)n

(bc/d)nψ d–,

φ+=

(de/bc;q)n–

(qe;q)n–

(qbc/d)n–ψ a+,

φ+ a–

=(de/bc;q)n (qe;q)n

(qbc/d)nψ.

(.)

Thus, from (.), changingφintoψ, and using (.) we get

d( –a)

(qe;q)n–

(de/bc;q)n–

(d/qbc)n–φ+–

(qe;q)n

(de/bc;q)n

(d/qbc)nφ+ a–

–a( –d)

(e;q)n

(de/bc;q)n

(d/bc)nφ– (qe;q)n (de/bc;q)n

(d/qbc)nφ+ a–

= .

After some simpliﬁcation this yields

a( –d)( –e)φ– (a–e)(a–d)φ+ a–

– ( –a)(qabc–de)φ+= . (.)

From the symmetries of theφ, we have

b( –d)( –e)φ– (b–e)(b–d)φ+ b–

– ( –b)(qabc–de)φ+= . (.)

Using (.), (.), and (.), we obtain the following contiguous relations:

ab(a–b)( –c)φ+b(a–e)(a–d)φ a––φ

–a(b–e)(b–d)φ b––φ= , (.)

qa( –b)( –c)φ+q(a–e)(a–d)φ a––φ

– ( –a)(qabc–de)φ a+–φ= , (.)

qb( –b)( –c)φ+q(b–e)(b–d)φ b––φ

– ( –b)(qabc–de)φ b+–φ= . (.)

Now, replacingbbyb/qin (.) andbbybqin (.) we get

(b–aq)φ b––φ+b( –a)φ–b( –a)φ a+,b–= , (.)

(a–qb)(ed–qabc)φ b+–φ+bq(a–e)(a–d)φ a–,b+

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Hence, combining (.) and (.) yields the three term contiguous relation

In this section, we state and prove the reduction formula for theq-integralIn,qstated in

the introduction. We start with the following result.

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Denoting theq-integral in the left-hand-side of (.) byJn, we obtain

Substituting (.) into (.), using (.), yields

In,q=

Using the transformation (.) yields the required result and completes the proof.

Corollary . Ifγ = then In,q vanishes for all values of n where n–β –δ andβ are

wherem,mare arbitrary nonnegative integers (see []), the proof follows directly from

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Watson remarkedInvanishes for odd values ofnin two special cases, (i)α=γandβ=δ

and (ii)α=βandγ =δ.

Now, we can derive the reduction formula forIn,q.

Theorem . The reduction formula satisﬁes a three term recurrence relations of In,q. More precisely,the following holds:

If

Wn=φ q–n,q–β–δ–α–γ+n–,q–β;q–β–δ,q–α–β;q,q

,

then

LnWn+–QnWn–+MnWn= , (.)

where

Ln= q–n–qθ q–n–qθ  –qθ+n qθ+n–q–n

,

Qn=  –q–n qθ+n–qθ qθ+n–qθ q–n–qθ+n+

,

Mn=qθ  –q–n

q–n qθ_–_qθ+n₊_q(θ+n)+₊_q(θ+n) _q–n_–_q–β

– []qθ+n _qθ+θ₊_q+θ₊_q+θ–n _{[] –}_q–n_–_qθ₊_q–n _q–n–β_–_qθ +q qθ+n_–_q–n _q+θ–β₊_qθ _qθ+n₊_q–n₊_qθ+θ _q–n₊_q(θ+n) + []qθ _qθ_–_q–β _q–n_–_qθ+n_,

andθ= –α–β,θ= –δ–β,θ= –α–β–δ–γ – .

Proof This result follows by applying Proposition . and using equation (.) with

b=q–β–δ–α–γ+n–, c=q–β, d=q–β–δ and e=q–α–β.

Recall that the littleq-Jacobi polynomials, see [], are deﬁned by

Pn(x;a,b;q) =φ q–n,abqn+;aq;q,qx

, (.)

and the formula

Pn(x;c,d;q) = n

k=

ak,nPk(x;a,b;q)

holds with

ak,n=Ckφ qk–n,cdqn+k+,aqk+;cqk+,abqk+;q,q

,

Ck= (–)kq

(q–n_,_aq_,_cdqn+_;_q₎

k

(q,cq,abqk+_;_q₎

k

.

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Remark . In (.), if we takea=q–β–_,_b₌_q–α–_,_c₌_q–β–δ–_and_d₌_q–α–γ–_{, we get}

a,n=_SnIn,q. Thus, the littleq-Jacobi polynomials and theq-integralsIn,qare related in the

following way:

Pn(x;c,d;q) =



Sn In,q+

n

k=

ak,nPk(x;a,b;q).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have made equal and signiﬁcant contributions in writing this paper. They read and approved the ﬁnal manuscript.

Author details

1_{Present address: Department of Mathematics, Faculty of Science, King Saud University, Riyadh, KSA.}2_{Permanent address:}

Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.3_{Department of Natural and Engineering}

Science, Faculty of Applied Studies and Community Service, King Saud University, Riyadh, KSA.

Acknowledgements

The authors would like to thank Prof. Mourad Ismail for pointing out the relation betweenIn,qand the littleq-Jacobi

polynomials. They are very grateful to the reviewers and editors for their suggestions and comments which improved ﬁnal version of this paper. This research is supported by the DSFP program of the King Saud University in Riyadh through grant DSFP/MATH 01.

Received: 25 October 2015 Accepted: 13 March 2016

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