Polynomial Generalizations of two-variable Ramanujan type identities

James Mc Laughlin; Andrew V. Sills

arXiv:1901.05328·math.NT·January 17, 2019·Electron. J. Comb.

Polynomial Generalizations of two-variable Ramanujan type identities

James Mc Laughlin, Andrew V. Sills

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TL;DR

This paper develops finite analogs of two-variable q-series identities from Ramanujan's lost notebook, expanding the understanding of these classical identities in a finite setting.

Contribution

It introduces polynomial generalizations of Ramanujan's two-variable identities, providing new finite analogs and extending their applicability.

Findings

01

Finite analogs of Ramanujan's identities are constructed.

02

The identities are expressed as polynomial generalizations.

03

Results deepen the understanding of q-series identities in finite forms.

Abstract

We provide finite analogs of a pair of two-variable $q$ -series identities from Ramanujan's lost notebook and a companion identity.

Equations93

j = 0 \sum \infty \frac{q ^{2 j^{2}} ( z q ; q ^{2} ) _{j} ( q / z ; q ^{2} ) _{j}}{( q ^{2} ; q ^{2} ) _{2 j}} = \frac{( z q ^{3} , q ^{3} / z , q ^{6} ; q ^{6} ) _{\infty}}{( q ^{2} ; q ^{2} ) _{\infty}},

j = 0 \sum \infty \frac{q ^{2 j^{2}} ( z q ; q ^{2} ) _{j} ( q / z ; q ^{2} ) _{j}}{( q ^{2} ; q ^{2} ) _{2 j}} = \frac{( z q ^{3} , q ^{3} / z , q ^{6} ; q ^{6} ) _{\infty}}{( q ^{2} ; q ^{2} ) _{\infty}},

(A; q)_{\infty} := (1 - A) (1 - A q) (1 - A q^{2}) \dots \mbox an d (A; q)_{n} := \frac{( A ; q ) _{\infty}}{( A q ^{n} ; q ) _{\infty}},

(A; q)_{\infty} := (1 - A) (1 - A q) (1 - A q^{2}) \dots \mbox an d (A; q)_{n} := \frac{( A ; q ) _{\infty}}{( A q ^{n} ; q ) _{\infty}},

(A_{1}, A_{2}, \dots, A_{r}; q)_{\infty} := (A_{1}; q)_{\infty} (A_{2}; q)_{\infty} \dots (A_{r}; q)_{\infty} .

(A_{1}, A_{2}, \dots, A_{r}; q)_{\infty} := (A_{1}; q)_{\infty} (A_{2}; q)_{\infty} \dots (A_{r}; q)_{\infty} .

j = 0 \sum \infty \frac{q ^{j (j + 1)} ( z ; q ) _{j} ( q / z ; q ) _{j + 1}}{( q ; q ) _{2 j + 1}} = \frac{( z q ^{2} , q / z , q ^{3} ; q ^{3} ) _{\infty}}{( q ; q ) _{\infty}} .

j = 0 \sum \infty \frac{q ^{j (j + 1)} ( z ; q ) _{j} ( q / z ; q ) _{j + 1}}{( q ; q ) _{2 j + 1}} = \frac{( z q ^{2} , q / z , q ^{3} ; q ^{3} ) _{\infty}}{( q ; q ) _{\infty}} .

j = 0 \sum \infty \frac{q ^{j^{2}} ( z q ; q ^{2} ) _{j} ( q / z ; q ^{2} ) _{j}}{( q ; q ^{2} ) _{j} ( q ^{4} ; q ^{4} ) _{j}} = \frac{( z q ^{2} , q ^{2} / z , q ^{4} ; q ^{4} ) _{\infty} ( - q ; q ^{2} ) _{\infty}}{( q ^{2} ; q ^{2} ) _{\infty}} .

j = 0 \sum \infty \frac{q ^{j^{2}} ( z q ; q ^{2} ) _{j} ( q / z ; q ^{2} ) _{j}}{( q ; q ^{2} ) _{j} ( q ^{4} ; q ^{4} ) _{j}} = \frac{( z q ^{2} , q ^{2} / z , q ^{4} ; q ^{4} ) _{\infty} ( - q ; q ^{2} ) _{\infty}}{( q ^{2} ; q ^{2} ) _{\infty}} .

j = 0 \sum \infty \frac{z ^{j} q ^{j^{2}}}{( q ; q ) _{j}} = \frac{1}{( z q ; q ) _{\infty}} n = 0 \sum \infty \frac{( - 1 ) ^{j} z ^{2 j} q ^{j (5 j - 1) /2} ( 1 - z q ^{2 j} ) ( z ; q ) _{j}}{( 1 - z ) ( q ; q ) _{j}},

j = 0 \sum \infty \frac{z ^{j} q ^{j^{2}}}{( q ; q ) _{j}} = \frac{1}{( z q ; q ) _{\infty}} n = 0 \sum \infty \frac{( - 1 ) ^{j} z ^{2 j} q ^{j (5 j - 1) /2} ( 1 - z q ^{2 j} ) ( z ; q ) _{j}}{( 1 - z ) ( q ; q ) _{j}},

j = 0 \sum \infty \frac{q ^{j^{2}}}{( q ; q ) _{j}} = \frac{1}{( q ; q ^{5} ) _{\infty} ( q ^{4} ; q ^{5} ) _{\infty}},

j = 0 \sum \infty \frac{q ^{j^{2}}}{( q ; q ) _{j}} = \frac{1}{( q ; q ^{5} ) _{\infty} ( q ^{4} ; q ^{5} ) _{\infty}},

j = 0 \sum \infty \frac{q ^{j (j + 1)}}{( q ; q ) _{j}} = \frac{1}{( q ^{2} ; q ^{5} ) _{\infty} ( q ^{3} ; q ^{5} ) _{\infty}},

j = 0 \sum \infty \frac{q ^{j (j + 1)}}{( q ; q ) _{j}} = \frac{1}{( q ^{2} ; q ^{5} ) _{\infty} ( q ^{3} ; q ^{5} ) _{\infty}},

\genfrac{[}{]}{0.0pt}{}{A}{B}_{q}:=\left\{\begin{array}[]{ll}\displaystyle{\frac{(q;q)_{A}}{(q;q)_{B}(q;q)_{A-B}}},&\mbox{ if $0\leq B\leq A$}\\ 0,&\mbox{otherwise}\end{array},\right.

\genfrac{[}{]}{0.0pt}{}{A}{B}_{q}:=\left\{\begin{array}[]{ll}\displaystyle{\frac{(q;q)_{A}}{(q;q)_{B}(q;q)_{A-B}}},&\mbox{ if $0\leq B\leq A$}\\ 0,&\mbox{otherwise}\end{array},\right.

\genfrac{[}{]}{0.0pt}{}{A}{B}_{q}^{*}:=\left\{\begin{array}[]{ll}1,&\mbox{ if $A=-1$ and $B=0$,}\\ \genfrac{[}{]}{0.0pt}{}{A}{B}_{q},&\mbox{otherwise}\end{array}.\right.

\genfrac{[}{]}{0.0pt}{}{A}{B}_{q}^{*}:=\left\{\begin{array}[]{ll}1,&\mbox{ if $A=-1$ and $B=0$,}\\ \genfrac{[}{]}{0.0pt}{}{A}{B}_{q},&\mbox{otherwise}\end{array}.\right.

T_{0} (L, A; q) := r = 0 \sum L (- 1)^{r} [r L]_{q^{2}} [L - A - r 2 L - 2 r]_{q} .

T_{0} (L, A; q) := r = 0 \sum L (- 1)^{r} [r L]_{q^{2}} [L - A - r 2 L - 2 r]_{q} .

\left[\genfrac{}{}{0.0pt}{}{A}{B};q,z\right]:=\left\{\begin{array}[]{ll}0&\mbox{if $B<0$}\\ 1&\mbox{if $B=0$ or $B=A$}\\ \sum_{h=0}^{B}z^{h}\genfrac{[}{]}{0.0pt}{}{A-B+h-1}{h}_{q}&\mbox{if $0<B<A$}\\ (zq^{A-B};q)_{B-A}&\mbox{if $B>A$.}\end{array}\right.

\left[\genfrac{}{}{0.0pt}{}{A}{B};q,z\right]:=\left\{\begin{array}[]{ll}0&\mbox{if $B<0$}\\ 1&\mbox{if $B=0$ or $B=A$}\\ \sum_{h=0}^{B}z^{h}\genfrac{[}{]}{0.0pt}{}{A-B+h-1}{h}_{q}&\mbox{if $0<B<A$}\\ (zq^{A-B};q)_{B-A}&\mbox{if $B>A$.}\end{array}\right.

j = 0 \sum n z^{j} q^{j^{2}} [j n]_{q} = j = 0 \sum n (- 1)^{j} q^{j (5 j - 1) /2} (1 - z q^{2 j}) [j n]_{q} \frac{1}{( z q ^{j} ; q ) _{n + 1}}

j = 0 \sum n z^{j} q^{j^{2}} [j n]_{q} = j = 0 \sum n (- 1)^{j} q^{j (5 j - 1) /2} (1 - z q^{2 j}) [j n]_{q} \frac{1}{( z q ^{j} ; q ) _{n + 1}}

j = 0 \sum n z^{n} q^{n^{2}} = 0 \leq 2 j \leq n \sum (- 1)^{j} z^{2 j} q^{j (5 j - 1) /2} (1 - z q^{2 j}) [j n]_{q} [j n - j]_{q} (q; q)_{j} \times \frac{( z ^{2} q ^{n + 2 j + 1} ; q ) _{n - 2 j}}{( z q ^{j} ; q ) _{n - j + 1}}

j = 0 \sum n z^{n} q^{n^{2}} = 0 \leq 2 j \leq n \sum (- 1)^{j} z^{2 j} q^{j (5 j - 1) /2} (1 - z q^{2 j}) [j n]_{q} [j n - j]_{q} (q; q)_{j} \times \frac{( z ^{2} q ^{n + 2 j + 1} ; q ) _{n - 2 j}}{( z q ^{j} ; q ) _{n - j + 1}}

j = 0 \sum n z^{j} q^{j^{2}} [j n; q, q] = 0 \leq 2 j \leq n \sum (- 1)^{j} z^{2 j} q^{j (5 j - 1) /2} [j n; q, q] [n - 2 j 2 n + 1 - 2 j; q, z q^{j}] - 0 \leq 2 j \leq n - 1 \sum (- 1)^{j} z^{2 j + 1} q^{j (5 j + 3) /2} [j n; q, q] [n - 2 j - 1 2 n - 2 j; q, z q^{j}],

j = 0 \sum n z^{j} q^{j^{2}} [j n; q, q] = 0 \leq 2 j \leq n \sum (- 1)^{j} z^{2 j} q^{j (5 j - 1) /2} [j n; q, q] [n - 2 j 2 n + 1 - 2 j; q, z q^{j}] - 0 \leq 2 j \leq n - 1 \sum (- 1)^{j} z^{2 j + 1} q^{j (5 j + 3) /2} [j n; q, q] [n - 2 j - 1 2 n - 2 j; q, z q^{j}],

j = 0 \sum ⌊ n /2 ⌋ h = 0 \sum j i = 0 \sum j (- 1)^{h + i} z^{h - i} q^{h^{2} + i^{2} + 2 j^{2}} [h j]_{q^{2}} [i j]_{q^{2}} [2 j j + ⌊ \frac{n - h - i}{2} ⌋]_{q^{2}} = j = - \infty \sum \infty (- 1)^{j} z^{j} q^{3 j^{2}} [⌊ \frac{n + 3 j - 1}{2} ⌋ n - 1]_{q^{2}} + ϵ_{n} (z, q),

j = 0 \sum ⌊ n /2 ⌋ h = 0 \sum j i = 0 \sum j (- 1)^{h + i} z^{h - i} q^{h^{2} + i^{2} + 2 j^{2}} [h j]_{q^{2}} [i j]_{q^{2}} [2 j j + ⌊ \frac{n - h - i}{2} ⌋]_{q^{2}} = j = - \infty \sum \infty (- 1)^{j} z^{j} q^{3 j^{2}} [⌊ \frac{n + 3 j - 1}{2} ⌋ n - 1]_{q^{2}} + ϵ_{n} (z, q),

\epsilon_{n}(z,q)\\ =\left\{\begin{array}[]{ll}\displaystyle{\sum_{j=-\infty}^{\infty}z^{2j}q^{12j^{2}+6j+n}\genfrac{[}{]}{0.0pt}{}{n-1}{\frac{n+6j}{2}}_{q^{2}}^{*}}&\mbox{if $2\mid n$}\\ \displaystyle{-\sum_{j=-\infty}^{\infty}z^{2j-1}q^{12j^{2}-6j+n}\genfrac{[}{]}{0.0pt}{}{n-1}{\frac{n+6j-3}{2}}_{q^{2}}^{*}}&\mbox{if $2\nmid n$}\end{array}\right.

\epsilon_{n}(z,q)\\ =\left\{\begin{array}[]{ll}\displaystyle{\sum_{j=-\infty}^{\infty}z^{2j}q^{12j^{2}+6j+n}\genfrac{[}{]}{0.0pt}{}{n-1}{\frac{n+6j}{2}}_{q^{2}}^{*}}&\mbox{if $2\mid n$}\\ \displaystyle{-\sum_{j=-\infty}^{\infty}z^{2j-1}q^{12j^{2}-6j+n}\genfrac{[}{]}{0.0pt}{}{n-1}{\frac{n+6j-3}{2}}_{q^{2}}^{*}}&\mbox{if $2\nmid n$}\end{array}\right.

j = 0 \sum ⌊ n /2 ⌋ h = 0 \sum j i = 0 \sum j + 1 (- 1)^{h + i} z^{h - i} q^{(2 h) + (2 i + 1) + j (j + 1)} [h j]_{q} [i j + 1]_{q} [2 j + 1 j + 1 + ⌊ \frac{n - h - i}{2} ⌋]_{q} = j = - \infty \sum \infty (- 1)^{j} z^{j} q^{j (3 j + 1) /2} [⌊ \frac{n + 3 j + 2}{2} ⌋ n]_{q} + ϵ_{n} (z, q),

j = 0 \sum ⌊ n /2 ⌋ h = 0 \sum j i = 0 \sum j + 1 (- 1)^{h + i} z^{h - i} q^{(2 h) + (2 i + 1) + j (j + 1)} [h j]_{q} [i j + 1]_{q} [2 j + 1 j + 1 + ⌊ \frac{n - h - i}{2} ⌋]_{q} = j = - \infty \sum \infty (- 1)^{j} z^{j} q^{j (3 j + 1) /2} [⌊ \frac{n + 3 j + 2}{2} ⌋ n]_{q} + ϵ_{n} (z, q),

\epsilon_{n}(z,q)=\left\{\begin{array}[]{ll}\displaystyle{\sum_{j=-\infty}^{\infty}z^{2j}q^{6j^{2}-2j+n/2}\genfrac{[}{]}{0.0pt}{}{n}{\frac{n}{2}+3j}_{q}}&\mbox{if $2\mid n$},\\ \displaystyle{-\sum_{j=-\infty}^{\infty}z^{2j+1}q^{6j^{2}+4j+\frac{1}{2}+\frac{n}{2}}\genfrac{[}{]}{0.0pt}{}{n}{\frac{n+6j+3}{2}}_{q}}&\mbox{if $2\nmid n$}.\end{array}\right.

\epsilon_{n}(z,q)=\left\{\begin{array}[]{ll}\displaystyle{\sum_{j=-\infty}^{\infty}z^{2j}q^{6j^{2}-2j+n/2}\genfrac{[}{]}{0.0pt}{}{n}{\frac{n}{2}+3j}_{q}}&\mbox{if $2\mid n$},\\ \displaystyle{-\sum_{j=-\infty}^{\infty}z^{2j+1}q^{6j^{2}+4j+\frac{1}{2}+\frac{n}{2}}\genfrac{[}{]}{0.0pt}{}{n}{\frac{n+6j+3}{2}}_{q}}&\mbox{if $2\nmid n$}.\end{array}\right.

\sum_{j=0}^{n}\ \sum_{h=0}^{j}\sum_{i=0}^{j}\sum_{\ell=0}^{n-h-i}(-1)^{h+i+\ell}z^{h-i}q^{h^{2}+i^{2}+j^{2}+2\ell}\genfrac{[}{]}{0.0pt}{}{j}{h}_{q^{2}}\genfrac{[}{]}{0.0pt}{}{j}{i}_{q^{2}}\genfrac{[}{]}{0.0pt}{}{j+\ell-1}{\ell}_{q^{2}}^{*}\\ \times\genfrac{[}{]}{0.0pt}{}{n-h-i+j-\ell}{2j}_{q}\\ =\sum_{j=-\infty}^{\infty}(-1)^{j}z^{j}q^{2j^{2}}\bigg{(}T_{0}(n,2j;q)+T_{0}(n-1,2j;q)\bigg{)}

\sum_{j=0}^{n}\ \sum_{h=0}^{j}\sum_{i=0}^{j}\sum_{\ell=0}^{n-h-i}(-1)^{h+i+\ell}z^{h-i}q^{h^{2}+i^{2}+j^{2}+2\ell}\genfrac{[}{]}{0.0pt}{}{j}{h}_{q^{2}}\genfrac{[}{]}{0.0pt}{}{j}{i}_{q^{2}}\genfrac{[}{]}{0.0pt}{}{j+\ell-1}{\ell}_{q^{2}}^{*}\\ \times\genfrac{[}{]}{0.0pt}{}{n-h-i+j-\ell}{2j}_{q}\\ =\sum_{j=-\infty}^{\infty}(-1)^{j}z^{j}q^{2j^{2}}\bigg{(}T_{0}(n,2j;q)+T_{0}(n-1,2j;q)\bigg{)}

(t; q)_{j} = h = 0 \sum j (- 1)^{h} t^{h} q^{h (h - 1) /2} [h j]_{q}

(t; q)_{j} = h = 0 \sum j (- 1)^{h} t^{h} q^{h (h - 1) /2} [h j]_{q}

\frac{1}{( t ; q ) _{j}} = h = 0 \sum \infty t^{h} [h h + j - 1]_{q}^{*}

\frac{1}{( t ; q ) _{j}} = h = 0 \sum \infty t^{h} [h h + j - 1]_{q}^{*}

ϕ (z, q) := j = 0 \sum \infty \frac{q ^{2 j^{2}} ( z q ; q ^{2} ) _{j} ( q / z ; q ^{2} ) _{j}}{( q ^{2} ; q ^{2} ) _{2 j}} .

ϕ (z, q) := j = 0 \sum \infty \frac{q ^{2 j^{2}} ( z q ; q ^{2} ) _{j} ( q / z ; q ^{2} ) _{j}}{( q ^{2} ; q ^{2} ) _{2 j}} .

f (t) := f (t; z, q) := j = 0 \sum \infty \frac{t ^{2 j} ( 1 + t ) q ^{2 j^{2}} ( t z q ; q ^{2} ) _{j} ( tq / z ; q ^{2} ) _{j}}{( t ^{2} ; q ^{2} ) _{2 j + 1}},

f (t) := f (t; z, q) := j = 0 \sum \infty \frac{t ^{2 j} ( 1 + t ) q ^{2 j^{2}} ( t z q ; q ^{2} ) _{j} ( tq / z ; q ^{2} ) _{j}}{( t ^{2} ; q ^{2} ) _{2 j + 1}},

f (t) = n = 0 \sum \infty P_{n} (z, q) t^{n} .

f (t) = n = 0 \sum \infty P_{n} (z, q) t^{n} .

t \to 1 - lim (1 - t) f (t; z, q) = ϕ (z, q)

t \to 1 - lim (1 - t) f (t; z, q) = ϕ (z, q)

n \to \infty lim P_{n} (z, q) = ϕ (z, q) .

n \to \infty lim P_{n} (z, q) = ϕ (z, q) .

f (t)

f (t)

= \frac{1}{1 - t} + j = 1 \sum \infty \frac{t ^{2 j} ( 1 + t ) q ^{2 j^{2}} ( t z q ; q ^{2} ) _{j} ( tq / z ; q ^{2} ) _{j}}{( t ^{2} ; q ^{2} ) _{2 j + 1}}

= \frac{1}{1 - t} + j = 0 \sum \infty \frac{t ^{2 j + 2} q ^{2 j^{2} + 4 j + 2} ( t z q ; q ^{2} ) _{j + 1} ( tq / z ; q ^{2} ) _{j + 1}}{( t ^{2} ; q ^{2} ) _{2 j + 3}}

= \frac{1}{1 - t} + \frac{t ^{2} q ^{2} ( 1 - t z q ) ( 1 - tq / z )}{( 1 - t ^{2} q ^{2} ) ( 1 + t q ^{2} ) ( 1 - t )} f (t q^{2}) .

(1 - t^{2} q^{2}) (1 + t q^{2}) (1 - t) f (t) = (1 - t^{2} q^{2}) (1 + t q^{2}) + t^{2} q^{2} (1 - t z q) (1 - tq / z) f (t q^{2}),

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TopicsAdvanced Mathematical Identities · Analytic Number Theory Research · Molecular spectroscopy and chirality

Full text

Polynomial Generalizations of two-variable Ramanujan type identities

James McLaughlin

[email protected]

Department of Mathematics, West Chester University, West Chester, Pennsylvania

and

Andrew V. Sills

[email protected]

Department of Mathematical Sciences, Georgia Southern University, Statesboro, Georgia 30460-8093

Dedicated to Doron Zeilberger on the occasion of his sixtieth birthday.

Abstract.

We provide finite analogs of a pair of two-variable $q$ -series identities from Ramanujan’s lost notebook and a companion identity.

Key words and phrases:

Ramanujan lost notebook, Rogers-Ramanujan identities

2000 Mathematics Subject Classification:

Primary 11B65; Secondary 05A10

“The progress of mathematics can be viewed as progress from the infinite to the finite.” —Gian-Carlo Rota (1983)

1. Introduction

At the top of a page in the lost notebook [14, p. 33] (cf. [6, p. 99, Entry 5.3.1]):, Ramanujan recorded an identity equivalent to the following:

[TABLE]

where we are employing the standard notation for rising $q$ -factorials,

[TABLE]

and

[TABLE]

In a recent paper [11], we found a partner to (1.1) that Ramanujan appears to have missed:

[TABLE]

Later on the same page of the lost notebook, Ramanujan recorded [6, p. 103, Entry 5.3.5]

[TABLE]

For further discussion of these three identities, see [10].

*Remark**.*

Out of respect for Doron’s ultra-finitist philosophy, we deliberately refrain from stating conditions on $q$ and $z$ which imply analytic convergence of the infinite series and products in (1.1)–(1.3).

The preceding identities stand out among identities of Rogers-Ramanujan type because they are two-variable series-product identities. While Rogers-Ramanujan type identities admit two-variable generalizations, most lose the infinite product representation in the two-variable case.

For example, in the standard two variable generalization of the first Rogers-Ramanujan identity,

[TABLE]

the right hand side reduces to an infinite product only for certain particular values of $z$ , e.g. $z=1$ gives the first Rogers-Ramanujan identity,

[TABLE]

while $z=q$ gives the second Rogers-Ramanujan identity,

[TABLE]

after application of the Jacobi triple product identity [6, p. 17, Eq. (1.4.8)].

In [16, §3], the second author presented nontrivial polynomial generalizations of all 130 Rogers-Ramanujan type identities appearing in Slater’s paper [18]. All of Slater’s identities involved one variable only. Here, we demonstrate that the methods employed in [16] can be used to obtain polynomial generalizations of the rarer species of two-variable $q$ -series-product identities as well.

2. Polynomial Generalizations

Define the standard binomial coëfficient by

[TABLE]

and the modified $q$ -binomial coëfficient by

[TABLE]

In [4], Andrews and Baxter define several $q$ -analogs of trinomial coëfficients; we shall require one of them here:

[TABLE]

More recently, Andrews [3] introduced the following generalization of the $q$ -binomial coëfficient:

[TABLE]

The following polynomial generalizations of (1.4) are known:

[TABLE]

(see [1, 7, 9, 12]),

[TABLE]

[8, Eq. (3.5)], and

[TABLE]

[3, p. 41, Eq. (1.11)].

Andrews [3] notes that one of his motivations for introducing (2.3) is that both sides of the equation are clearly polynomials term by term, whereas this is not the case for the right hand sides of (2.1) and (2.2). The polynomial identities we introduce below also have this desirable feature.

Notice that in each of the identities below, the summands have finite support, and follow the natural bounds (i.e. each summation could be taken over all integers, and no nonzero terms would be added).

Identity 2.1 (Polynomial Generalization of (1.1)).

[TABLE]

where

[TABLE]

.

Identity 2.2 (Polynomial Generalization of (1.2)).

[TABLE]

where

[TABLE]

Identity 2.3 (Polynomial Generalization of (1.3)).

[TABLE]

3. Derivation and a method of proof

3.1. Identity 2.1

Recall the following consequences of the $q$ -binomial theorem:

[TABLE]

The derivation of (2.1) is via the method used for the derivations of polynomial versions of Rogers-Ramanujan type identities (in $q$ only) as introduced by Andrews [2, Chapter 9], and further explored by Santos [15] and the second author [16, 17]. We shall consider the details of (1.1) only; (1.2) and (1.3) may be treated analogously.

We begin with the left hand side of (1.1)

[TABLE]

Now define the following generalization of $\phi(z,q)$ :

[TABLE]

and let $P_{n}(z,q)$ be defined by

[TABLE]

Note that

[TABLE]

and

[TABLE]

Thus,

[TABLE]

which immediately implies

[TABLE]

Upon recalling that $f(t)=\sum_{n=0}^{\infty}P_{n}(z,q)t^{n}$ , and extracting the coëfficients of $t^{n}$ from (3.5), we find that the $P_{n}=P_{n}(z,q)$ satisfy the fourth order recurrence

[TABLE]

with initial conditions

[TABLE]

Thus we now have a full characterization of the $P_{n}(z,q)$ via a recurrence with initial conditions.

Next, we use $f(t)$ to derive the left hand side of (2.1).

[TABLE]

Compare coëfficients of $t^{n}$ in the extremes to find

[TABLE]

Next, after some inspired guesswork, (see [5, 16, 17] for details) we define the polynomials

[TABLE]

where

[TABLE]

Our goal is to show that the $P_{n}(z,q)$ and $Q_{n}(z,q)$ are in fact one and the same, thus giving us (2.1). We would like to use a computer implementation of the $q$ -Zeilberger algorithm [13, 19, 20, 21, 22] to simply show that the $Q_{n}$ satisfy the recurrence (3.6), and then upon checking that the $Q_{n}$ satisfy the initial conditions (3.7), we would be done. Unfortunately, the implementations of the $q$ -Zeilberger algorithm currently available do not allow for direct input of summands as complex as those under consideration here. And the corresponding certificate function would likely be rather horrendous. Further, it is unlikely that the $q$ -Zeilberger algorithm would produce a minimal recurrence for the $Q_{n}$ . So, the traditional automated proof would require a certain amount of pre-processing and post-processing.

3.2. Identity 2.2

The derivation is analogous to that of Identity 2.1. The recurrence and initial conditions are

[TABLE]

with

[TABLE]

3.3. Identity 2.3

The recurrence and initial conditions are

[TABLE]

with

[TABLE]

4. Challenge

We leave it as a challenge to produce automated proofs for Identities 2.1– 2.3.

Acknowledgments

Many thanks to Doron Zeilberger for revolutionizing the way we approach the discovery and proof of identities, especially those of the hypergeometric and $q$ -hypergeometric type.

Bibliography22

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] G. E. Andrews, Problem 74-12, SIAM Review 16 (1974).
2[2] G. E. Andrews, q 𝑞 q -Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra , C.B.M.S. Regional Conference Series in Math, No. 66, American Math. Soc. Providence, 1986.
3[3] G. E. Andrews, a 𝑎 a -Gaussian polynomials and finite Rogers-Ramanujan identities, in: Theory and Applications of Special Functions: a Volume Dedicated to Mizan Rahman , M. Ismail and E. Koelink, eds., 39–60. Springer, New York, 2005.
4[4] G. E. Andrews and R. J. Baxter, Lattice gas generalization of the hard hexagon model III: q 𝑞 q -trinomial coefficients, J. Statist. Phys 47 (1987) 298–330.
5[5] G. E. Andrews and R. J. Baxter, Scratchpad explorations for elliptic theta functions, In: D. V. Chudnovsky and R. D. Jenks (eds.), Computers in Mathematics. Lecture Notes in Pure and Applied Mathematics , vol. 125., pp. 17–33, Marcel Dekker, Inc., 1990.
6[6] G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook, Part II , Springer, 2009.
7[7] D. M. Bressoud, Solution to problem 74-12, SIAM Review 23 (1981) 101–104.
8[8] D. M. Bressoud, Some identities for terminating q 𝑞 q -series, Math Proc. Cambridge Phil. Soc. 89 (1981) 211–223.

TL;DR

Contribution

Findings

Abstract

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Polynomial Generalizations of two-variable Ramanujan type identities

Abstract.

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2000 Mathematics Subject Classification:

1. Introduction

Remark*.*

2. Polynomial Generalizations

Identity 2.1** (Polynomial Generalization of (1.1)).**

Identity 2.2** (Polynomial Generalization of (1.2)).**

Identity 2.3** (Polynomial Generalization of (1.3)).**

3. Derivation and a method of proof

3.1. Identity 2.1

3.2. Identity 2.2

3.3. Identity 2.3

4. Challenge

Acknowledgments

*Remark**.*

Identity 2.1 (Polynomial Generalization of (1.1)).

Identity 2.2 (Polynomial Generalization of (1.2)).

Identity 2.3 (Polynomial Generalization of (1.3)).