Inversion techniques and combinatorial identities. Jackson’s <span class="mathjax-formula">$q$</span>-analogue of the Dougall-Dixon theorem and the dual formulae

C ^OMPOSITIO M ^ATHEMATICA

W ENCHANG C HU

Inversion techniques and combinatorial identities.

Jackson’s q-analogue of the Dougall-Dixon theorem and the dual formulae

Compositio Mathematica, tome 95, n

^o

1 (1995), p. 43-68

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43

Inversion

techniques

^and combinatorial identities -

Jackson’s q-analogue

^{of the}

Dougall-Dixon

^{Theorem and}

the dual formulae

WENCHANG CHU

Institute of Systems Science, Academia Sinica, Beijing ¹⁰⁰⁰⁸⁰ Received 6 July 1993; accepted in final form 9 December 1993

Abstract. As the common extension of theorems due to Gould and Hsu (1973) ^{and Carlitz} (1973),

a general pair ^of reciprocal ^relations is established. By ^{means of the} embedding machinery, ^{it is}

used to demonstrate that numerous q-hypergeometric identities are the dual relations of _q- Saalschutz and Dougall-Dixon formulae. This fact serves as a natural reason for the existence of strange evaluations of basic hypergeometric ^series.

1. Introduction As

usual,

^denote

by

the binomial coefficient and the Gaussian

q-binomial

coefficient _respec-

tively.

^{For the two}

complex

sequences

{ak}

^and

{bk},

^{define the}

03C8-poly-

nomials

by

Then there exists a useful

pair

^of inverse series relations

which was discovered

by

^{Gould and Hsu}

[18]

in 1973. At the same

time,

Partially supported by ^{Alexander von} Humboldt Foundation (Germany), ^Italian Consiglio

Nazionale Delle Ricerche (CNR) ^{and NSF} (Chinese), Youth-Grant no. 1990-1033.

Compositio Mathematica 95: 43-68, 1995.

(Ç) 1995 Kluwer Academic Publishers. Printed in the Netherlands.

Carlitz found its

q-analogue

^which may be restated as follows:

Both

(1.2)

^and

(1.3)

^{have been}

neglected,

^even

though special

^{versions were}

rediscovered

by

^Andrews

([1], Bailey pair)

and Gessel and Stanton

([15,16], Lagrange inversions)

^{and used to} prove

hypergeometric

^{formulas. Their}

efficiency

^for

confirming

combinatorial

identities, through

^the

embedding machinery,

^was

recognized

ⁱⁿ

[7,9].

As a natural extension of Carlitz’

theorem,

^{a more}

general pair

^{of inverse}

series relations will be established in this paper. The

development

of inversion

technique

^{will be used to} demonstrate numerous strange

q-hypergeometric identities, by properly embedding

the variants of the

q-Saalschutz

theorem and Jackson’s

q-analogue

^of

Dougall-Dixon

theorem in the members of this new

pair

^of inversions. For

convenience,

^{these two}

q-series

identities _{may be}

displayed

^{as below:}

where the usual notation of

q-shifted-factorial,

Gaussian binomial

coefficients,

and basic

hypergeometric

series from the

monographs by Bailey [3], Gasper

and Rahman

[14]

and Slater

[24]

^{have been}

adopted

^{and the} fraction of

q-factorials

abbreviated as

45

The results exhibited in the present paper

together

^{with the}

previous

^one

[7]

demonstrate a remarkable fact that the most basic

hypergeometric

^identities

are the dual relations of

only

^three

q-formulas analogous

^{to those} named after

Chu-Vandermonde-Gauss,

Pfaff-Saalschutz and

Dougall-Dixon.

^{This serves as}

a natural reason for the existence of strange ^basic

hypergeometric

^relations.

2. A new

pair

^of

reciprocal

^relations

As an extension of Carlitz

theorem,

this section will establish a

general pair

of

reciprocal

relations contained in the

following

THEOREM. With the

03C8-polynomials defined by (1.1),

^{we have the inverse}

series relations:

provided

^{that the}

sequence-transforms

^{involved are}

non-singular,

^i.e.

03C8(03BBqn;

^{m +}

1), 03C8(q-n ; m + 1)

^and

(03BBqn ; q)m+1 do

^{not vanish}

for non-negative integers

^{m n.}

It is obvious that this

pair

of inversions will reduce ^to

(1.3)

^{when tends to}

infinity.

^{Similar to the} role of Carlitz

inversions, (2.1)

may be

used, systemati- cally,

^{to deal with} strange ^basic

hypergeometric

^series

through

^{the so-called}

embedding technique (or

^creative

telescoping)

^{mentioned in the} introduction.

The exhibition demonstrated in the next three sections will show that this

approach

^{can not}

only certify

^{the known}

q-hypergeometric formulas,

^{but also}

create several new strange evaluations.

Proof of the

^theorem.

Recalling

^the connection

[23,

p.

44-46]

^{between inverse}

series relations and matrix inverse

pairs,

^the

reciprocity

^between

(2.1a)

^and

(2.1b)

^is

equivalent

^to either of the

following orthogonal

^relations:

The former

simply

follows from

summand-splitting

and

diagonal-cancelling (but,

^{the dual}

orthogonality (2.2b)

^{is more difficult to}

confirm

directly).

^This

completes

^the

proof

^of the theorem. D The rotated version of the theorem may be stated as

COROLLARY. Assume the conditions

of

the theorem. The system

of equations

is

equivalent

^to the system

where N is an

arbitrary non-negative integer

^or

infinity.

One

pair

of bi-basic inverse relations due to Bressoud

[4]

^and

Gasper [12]

can also be

specified

^from

(2.1)

^and

(2.3) by defining 03C8-polynomials

^{to be}

shifted-factorials.

47

PROPOSITION. Let a and b be

complex

numbers such that

1-apXqY, 1 - bpXq-Y

and 1 ^-

qxalb differ from zero for non-negative integers

^{x and} y. Then we have the inverse series relations

and their rotated

forms

Based on the

theorem,

^the

remaining

part ^of the paper will

display

¹⁸

examples

which may be sketched as follows. If

(2.1a)

holds with the parameter 03BB and the sequences

{ai, bj and {f(k), g(k)l

^{under a certain}

specification (which correspond

^{to a known basic}

hypergeometric evaluation),

^{then we have}

(2.1b)

^{under the same}

specification (which

^{can be rewritten as another}

hypergeometric evaluation).

^Or

conversely,

^{this statement} is valid either

by interchanging (2.1 a)

^and

(2.1b).

3.

Embedding technique

^on balanced formulae The transforms

will be used to

demonstrate, by (2.4),

^{the dual relations between the}

balanced summations

(the

Saalschutzian

series)

^{and the} non-trivial _q-

hypergeometric

evaluations. All the balanced formulae

(cf. [8])

^involved

here may be

regarded

^{as the}

q-analogue

^of

Bailey’s

identities

[3,

p.

30]

^and

some of them have

appeared

^{in a}

slightly

different version

[21].

EXAMPLE 3.1. Saalschutz formula ^p the first non-trivial evaluation.

Rewrite the balanced formula

in the form of

(2.4b)

whose dual relation

is

equivalent

^to

a very-well poised

^evaluation

EXAMPLE 3.2. Balanced formula ^~ the second non-trivial evaluation.

Rewrite the balanced formula

1 nversion

techniques

and combinatorial identities 49 in the form of

(2.4b)

whose dual relation

is

equivalent

^to

a very-well poised

^evaluation

EXAMPLE 3.3. Balanced formulas thé third non-trivial evaluation.

Rewrite the balanced formula

in the form of (2.4b)

whose dual relation

is

equivalent

^to

a very-well poised

^evaluation

EXAMPLE 3.4. Balanced formula ~ the fourth non-trivial evaluation.

Rewrite the balanced formulae

in the form of

(2.4b)

51

whose dual relation

is

equivalent

^to

a very-well poised

^evaluation

EXAMPLE 3.5. Balanced formula ~ the fifth non-trivial evaluation.

Rewrite the balanced formula

in the form of

(2.4b)

whose dual relation

is

equivalent

^to

a very-well poised

^evaluation

In the similar _way, one can enumerate the dual relations of the so called bi-basic formulas

displayed

ⁱⁿ

[21] (see

^also

[14, §3.10]).

The details are

omitted here.

4.

Strange

évaluations associated with Jackson’s

q-Dougall-Dixon

^theorem

By

^{means of}

(3.0)

^{and transforms}

Jackson’s

q-Dougall-Dixon

^theorem

(1.5)

^{will be embedded in}

(2.1a)

^and generate,

through (2.1b),

^{the dual formulas. From this process, several}

unexpected

strange

hypergeometric

evaluations will be found.

Among

^the

examples displayed

^{in this}

section,

^{some of their inverse} relations

(i.e. Examples 4.2,

^{4.3 and}

4.6-4.9)

^{are not balanced}

by

^the

q-binomial coefficient n . Instead,

^{it is}

replaced by non-symmetric

^forms

n , n ]

^and

ⁿ

This is due to the

particular specification

^for

sequence

g(···).

^For

example,

^the

specification g(2k

⁺

1)

^{= 0 in}

Eq. (4.2b)

comes from the

special q-Dougall-Dixon

^formula

(4.2a).

EXAMPLE 4.1.

q-Dougall-Dixon

theorem - Gessel and Stanton’s evalu- ation.

The

specialized q-Dougall-Dixon

^formula

53

can be rewritten as

whose dual version

may be

expressed

^{as a} strange

terminating

^evaluation

which is due to Gessel and Stanton

[16, Eq. (1.4)].

^{For its}

non-terminating version,

^see

[12, Eq. (5.2)], [13, Eq. (5.1)]

^and

[14, Eq. (3.8.12)].

EXAMPLE 4.2.

q-Dougall-Dixon

théorème another Gessel and Stanton’s evaluation.

The

specialized q-Dougall-Dixon

^formula

can be rewritten _as, after

replacement q - q2

whose dual version

may be

expressed

^{as another} strange

terminating

^evaluation

which is also due to Gessel and Stanton

[16, Eq. (6.14)].

EXAMPLE 4.3.

q-Dougall-Dixon

theorem p the first new strange evaluation.

The

specialized q-Dougall-Dixon

^formula

55

can be rewritten _as, after

replacement q - q2

whose dual version

may be

expressed

^{as the first new} strange

terminating

evaluation in this section

EXAMPLE 4.4.

q-Dougall-Dixon

^{théorème the second new} strange ^evalu- ation.

The

specialized q-Dougall-Dixon

^formula

can be rewritten _as, after

replacement q2 ~

q

whose dual version

may be

expressed

^as the second new strange

terminating

evaluation in this section

EXAMPLE 4.5.

q-Dougall-Dixon théorème Gasper’s

strange evaluation.

The

specialized q-Dougall-Dixon

^formula

57

can be rewritten as

whose dual version

may be

expressed

^{as a} strange

terminating

^evaluation

which is due to

Gasper [12, Eq. (5.22)].

^{See also}

[13, Eq. (1.2)].

EXAMPLE 4.6.

q-Dougall-Dixon

^{theorem ~ the third new} strange evaluation.

The

specialized q-Dougall-Dixon

^formula

can be rewritten _as, after

replacement q ~ q3

whose dual version

(otherwise)

may be

expressed

^{as the third new} strange

terminating

evaluation in this section

(otherwise).

EXAMPLE 4.7.

q-Dougall-Dixon

theorem«the fourth ^new strange ^evalu- ation.

The

specialized q-Dougall-Dixon

^formula

59

can be rewritten as, after

replacement q - q4

whose dual version

(otherwise)

may be

expressed

^{as the fourth new} strange

terminating

evaluation in this section

(otherwise).

EXAMPLE 4.8.

q-Dougall-Dixon

theorem ~ the fifth new strange evaluation.

The

specialized q-Dougall-Dixon

^formula

can be rewritten _as, after

replacement q - q2

whose dual version

may be

expressed

^{as the fifth new} strange

terminating

evaluation in this section

61 EXAMPLE 4.9.

q-Dougall-Dixon

^{theorem ~ the sixth new} strange ^evaluati(

The

specialized q-Dougall-Dixon

^formula

can be rewritten _as, after

replacement

q ~

q3

whose dual version

(otherwise)

may be

expressed

^{as the last new} strange

terminating

evaluation in this section

(otherwise).

REMARK.

Similarly,

^{one can show that Jackson’s}

q-Dougall-Dixon

^for-

mula

(1.5)

with five free parameters, ^is

self-reciprocal,

i.e. the dual relation has

exactly

^{the same formation as the}

original

^{one under the} parameter

replacement.

^In

addition,

^{for a} very

special

^{case of}

q-Dougall-Dixon

theorem

its

resulting

dual relation is trivial.

5. Reversai

embeddings

^on

q-Dougall-Dixon

^theorem

In

general,

^a

terminating hypergeometric

^summation

Lk=O T(n, k)

⁼

S(n),

cannot be

telescoped

^{into one of the} relations stated in

(2.1).

^{But occa-}

sionally,

^{its reversal}

03A3nk = o T(n, n - k) = S(n)

may be restated as one member

of (2.1).

^{In this} case, the dual relation will create some

"mysterious-looking"

formulas.

By

^{means of}

(3.0), (4.0)

and transforms

some remarkable

examples

^are demonstrated as follows.

63 EXAMPLE 5.1.

q-Dougall-Dixon

^{theorem ~ one} very strange ^new evaluation.

The reversal of

q-Dougall-Dixon

^formula

can be

expressed

^{as, after}

adding

^some extra-zero terms

whose dual relation

can be reformulated in

q-series

which seems to be new.

EXAMPLE 5.2.

q-Dougall-Dixon theorem ~ Gasper

and Rahman’s two very strange evaluations.

For 03BB ⁼ 1 and

2,

the reversal of

q-Dougall-Dixon

^formula

can be

expressed

as, after

adding

^some extra-zero terms

whose dual relation

65

can be reformulated in

q-series

which

yields terminating Gasper

^{and Rahman’s formulas}

[13, Eqs. (1.8, 4.5)], respectively,

^{for 03BB = 1 and 2. See also}

[14, Eq. (3.8.18),

^Ex.

3.32].

EXAMPLE 5.3.

q-Dougall-Dixon

theorem ~ three very strange ^{new evalu-} ations.

For

03BB = 1, 2

^and

3,

^the reversal of

q-Dougall-Dixon

^formula

can be

expressed

as, after

adding

^some extra-zero terms

whose dual relation

can be reformulated in

q-series

which is the unified version of three new

terminating

strange evaluations

corresponding

^{to 03BB =}

1,

^{2 and 3.}

EXAMPLE 5.4.

q-Dougall-Dixon

^{theorem p two} very strange ^new evaluations.

For 03BB ⁼ 1 and

2,

^the reversal of

q-Dougall-Dixon

^formula

can be

expressed

^{as, after}

adding

^some extra-zero terms

67 whose dual relation

can be reformulated in

q-series

which is the unified version of two new

terminating

strange evaluations

corresponding

^{to 03BB = 1 and 2.}

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