Multi-sum-to-single-sum transformations: other base summations, an underlying WP-Bailey chain, and combinatorial proofs of the embedded Jacobi-type identities (Mc Laughlin 2016)

J. Mc Laughlin (Ramanujan J. 39 (2016) 545-565) proved two general transformations that reduce basic hypergeometric multi-sums of arbitrary depth, carrying an arbitrary input sequence g(j), to an infinite q-product times a single sum, the iteration deriving ultimately from the q-Gauss summation; he also recast one of them as a Bailey-type transformation (Theorem 6) between sequences linked by a one-parameter convolution. The paper's concluding remarks pose three open questions which together ask for a complete structural theory of this transformation method: (1) do analogous multi-sum-to-single-sum transformations derive from classical summation formulae other than the q-Gauss sum and the q-Pfaff-Saalschütz sum, the two bases already exploited respectively by Mc Laughlin and by Andrews and Chu; (2) is the Theorem-6 transformation a limiting case of a full, presently unknown WP-Bailey chain; (3) do the multi-sum identities (13), (14), (16) of the paper, which embed Jacobi's identity and Andrews-Gordon-type identities in infinite families, admit combinatorial proofs or interpretations.

Background

Fix a base $q$ with $|q|<1$ and write $(a;q)_n=\prod_{j=0}^{n-1}(1-aq^j)$ and $(a;q)_\infty=\prod_{j\ge0}(1-aq^j)$ for the q-shifted factorials. A basic hypergeometric series ${}_r\phi_s$ is a q-series in which the ratio of consecutive summands is a rational function of $q^n$; the summation and transformation formulae satisfied by these series form the classical layer of q-series theory on which the present problem sits. Two classical summations matter here. The nonterminating q-Gauss summation evaluates a ${}_2\phi_1$ series in closed product form,

$$\sum_{n=0}^{\infty}\frac{(a,b;q)_n}{(c,q;q)_n}\left(\frac{c}{ab}\right)^n=\frac{(c/a,c/b;q)_\infty}{(c,c/ab;q)_\infty},$$

and the q-Pfaff-Saalschütz summation evaluates a certain terminating ${}_3\phi_2$ series (terminating: the sum is finite because a numerator parameter is a non-positive integer) as a ratio of q-shifted factorials.

By a multi-sum-to-single-sum transformation one means an identity that takes a nested multiple sum (depth $k$, indices $m_1\ge m_2\ge\cdots\ge m_k\ge0$) whose summand carries an arbitrary input sequence $\{g(j)\}$ and reduces it to an infinite q-product times a single sum over $g$. Two classical sources of such transformations are known. First, the Bailey machinery: a Bailey pair is a pair of sequences $(\alpha_n,\beta_n)$ linked by a convolution relation, and a Bailey chain is a construction producing a new Bailey pair from an old one, so iterating the associated Bailey transform converts the pair relation into multi-sum identities of arbitrary depth. The well-poised (WP) analogue of Andrews replaces the pair relation by

$$\beta_n(a,\kappa,q)=\sum_{j=0}^{n}\frac{(\kappa/a;q)_{n-j}(\kappa;q)_{n+j}}{(q;q)_{n-j}(aq;q)_{n+j}}\alpha_j(a,\kappa,q),$$

and iterating WP-Bailey chains similarly generates very-well-poised multi-sum identities ("well-poised" refers to a symmetric pattern of parameters in the underlying series). Second, direct multiple q-series work: Andrews' Lusztig-Macdonald-Wall paper and his multiple Rogers-Ramanujan-type identities, and Chu's "Saalschütz chain reactions", iterate the q-Pfaff-Saalschütz summation.

Mc Laughlin's 2016 paper added a third, cleaner source. Seeking an analytic proof of a double-sum identity for which Pak's survey on partition bijections had asked for a combinatorial proof, he found the transformation $\sum_{m\ge n\ge0}\frac{(a;q)_m(b;q)_n(c/b;q)_{m-n}}{(c;q)_m(q;q)_n(a;q)_{m-n}}(c/ab)^n g(m-n)=\frac{(c/a,c/b;q)_\infty}{(c,c/ab;q)_\infty}\sum_{k\ge0}g(k)$, with an arbitrary sequence $g$ on both sides, proved by a single application of the q-Gauss summation, and then iterated it to arbitrary depth:

$$\sum_{\vec m}\prod_{j=1}^{k}\frac{(a_j;q)_{m_j}(c_j/b_j;q)_{m_{j+1}}(b_j;q)_{m_j-m_{j+1}}}{(c_j;q)_{m_j}(a_j;q)_{m_{j+1}}(q;q)_{m_j-m_{j+1}}}\left(\frac{c_j}{a_jb_j}\right)^{m_j-m_{j+1}}g(m_{k+1})=\prod_{j=1}^{k}\frac{(c_j/a_j,c_j/b_j;q)_\infty}{(c_j,c_j/a_jb_j;q)_\infty}\sum_{j=0}^{\infty}g(j),$$

summed over $m_1\ge\cdots\ge m_{k+1}\ge0$. Applications include summation formulae for continuous q-ultraspherical polynomials (a family in the q-Askey scheme of orthogonal polynomials) and multi-sums collapsing to infinite products, notably the families

$$\sum_{\vec m}\frac{q^{m_1(m_1-m_2)+m_2(m_2-m_3)+\cdots+m_{k-1}(m_{k-1}-m_k)+m_k^2}}{(q;q)_{m_1}(q;q)_{m_1-m_2}\cdots(q;q)_{m_{k-1}}(q;q)_{m_{k-1}-m_k}(q;q)_{m_k}^2}=\frac{1}{(q;q)_\infty^k}$$

and

$$\sum_{\vec m}\frac{q^{k[m_1(m_1-m_2)+\cdots+m_{k-1}(m_{k-1}-m_k)+m_k^2-m_1]+m_1+m_2+\cdots+m_k}}{(q^k;q^k)_{m_k}\prod_{i=1}^{k-1}(q^k;q^k)_{m_i-m_{i+1}}\prod_{i=1}^{k}(q^i;q^k)_{m_i}}=\frac{1}{(q;q)_\infty},$$

equations (13) and (14) of the paper, whose $k=1$ cases are Jacobi's identity $\sum_{n\ge0}q^{n^2}/(q;q)_n^2=1/(q;q)_\infty$ (combinatorially the generating function for partitions, with the classical bijective proof via Durfee squares, the largest square in a Ferrers diagram (the standard dot-diagram of a partition)), together with a bilateral family, equation (16) of the paper, valid for integers $k\ge1$, $p\ge3$ and $1\le i\le p/2$, and summed over $m_1\ge\cdots\ge m_k\ge0$ with $m_k\ge m_{k+1}>-\infty$:

$$\sum_{\vec m}\frac{q^{k[m_1(m_1-m_2)+m_2(m_2-m_3)+\cdots+m_k(m_k-m_{k+1})-m_1]+m_1+m_2+\cdots+m_k}}{\prod_{j=1}^{k}(q^k;q^k)_{m_j-m_{j+1}}\prod_{j=1}^{k}(q^j;q^k)_{m_j}}\,(q^{p/2})^{m_{k+1}^2}(-q^{p/2-i})^{m_{k+1}}=\frac{(q^i,q^{p-i},q^p;q^p)_\infty}{(q;q)_\infty},$$

whose right side coincides, for odd $p$, with the product side of the Andrews-Gordon identities (the standard multi-sum generalizations of the Rogers-Ramanujan identities) at modulus $p$. Finally, the double-sum theorem recast in terms of a restricted Bailey pair $\beta_m=\sum_{n=0}^m\frac{(b;q)_{m-n}}{(q;q)_{m-n}}\alpha_n$ becomes the Bailey-type transformation (Theorem 6 of the paper)

$$\sum_{m=0}^{\infty}\frac{(a;q)_m}{(c;q)_m}\left(\frac{c}{ab}\right)^m\beta_m=\frac{(c/a,c/b;q)_\infty}{(c,c/ab;q)_\infty}\sum_{k=0}^{\infty}\frac{(a;q)_k}{(c/b;q)_k}\left(\frac{c}{ab}\right)^k\alpha_k.$$

The restricted pair is exactly what one obtains from Andrews' WP-Bailey pair above by replacing $\kappa$ with $a\kappa$, letting $a\to0$, and setting $\kappa=b$; yet, as the paper observes, it does not appear that the Theorem-6 transformation follows upon making the same substitutions in any of the existing WP-Bailey chains — the only such chain containing free parameters different from $a$ and $\kappa$ is Andrews' first WP-Bailey chain, and the paper verifies that this chain degenerates to a trivial identity under the limit — so the transformation appears to be only a "shadow" of an undiscovered WP-Bailey chain. The paper's concluding remarks then pose the three questions that constitute the present problem: (i) do analogous multi-sum-to-single-sum transformations derive from classical summation formulae other than the q-Gauss sum (this paper's base) and the q-Pfaff-Saalschütz sum (the base of Andrews' and Chu's transformations); (ii) does the Theorem-6 transformation derive from some as-yet-undiscovered WP-Bailey chain under the substitution $\kappa\to\kappa a$, $a\to0$; (iii) are there combinatorial proofs of the identities (13), (14) and (16).

Problem Statement

Resolve the three linked open questions posed in the concluding remarks of Mc Laughlin (Ramanujan J. 39 (2016) 545-565), which together ask for a complete structural account of the multi-sum-to-single-sum transformation method introduced in that paper. The common scientific objective is to determine the scope of the method (whether classical summations other than those already exploited generate such transformations), its structural origin (which well-poised Bailey-chain structure the resulting transformations reflect), and the combinatorial content of its consequences (which partition families the resulting identities count). Concretely:

(1) Other base summations. Do there exist transformations of the source's type, reducing nested basic hypergeometric multi-sums of arbitrary depth that involve an arbitrary input sequence $\{g(j)\}$ to an infinite q-product times a single sum over the input sequence, that derive ultimately from classical summation formulae other than the q-Gauss summation (the source's base) and the q-Pfaff-Saalschütz summation (the base of Andrews' and Chu's transformations)? Candidate bases include the remaining classical summations of basic hypergeometric series, for example the q-Dixon, q-Whipple, or Jackson q-Dougall summations. An answer must exhibit each claimed transformation, with full convergence conditions and its derivation from the named summation. Exhibiting at least one such transformation for at least one base other than the q-Gauss and q-Pfaff-Saalschütz summations resolves this sub-question, provided the transformation is non-degenerate — that is, not obtainable as a limiting case of the source's q-Gauss-based transformation or of the q-Pfaff-Saalschütz-based transformations of Andrews and Chu.

(2) The WP-Bailey chain behind Theorem 6. Does the Bailey-type transformation stated in the Background (Theorem 6 of the source) derive from a full, presently unknown WP-Bailey chain, after replacing $\kappa$ with $\kappa a$ and letting $a\to0$, so that the restricted pair $\beta_m=\sum_{n=0}^m\frac{(b;q)_{m-n}}{(q;q)_{m-n}}\alpha_n$ arises as the corresponding limit of Andrews' WP-Bailey pair? An answer must either exhibit such a WP-Bailey chain and derive the Theorem-6 transformation from it by the stated limit, showing that the chain is not a relabeling of the known chains (which, as the source observes, do not appear to yield the transformation under this limit), or prove that no such chain exists.

(3) Combinatorial proofs. Are there combinatorial proofs, or combinatorial interpretations, of the multi-sum identities at (13), (14) and (16) of the source, reproduced in the Background? The source conjectures an interpretation of (13) in terms of multipartitions with $k$ components (ordered $k$-tuples of partitions, whose generating function $1/(q;q)_\infty^k$ is exactly the right side of (13)) and of (14) in terms of $k$-modular partitions (a standard generalization of partitions in partition theory; the source leaves the precise correspondence open). An answer must define the combinatorial objects on both sides and give a weight-preserving bijection (or equivalent combinatorial argument) valid for all depths $k\ge2$ for each of the three families (13), (14) and (16); for the families (13) and (14) the $k=1$ case is Jacobi's identity with its classical Durfee-square proof, while the bilateral family (16) has no Jacobi specialization.

The problem is completely answered when each of (1)-(3) is resolved, each part by an explicit construction and proof or by a rigorous obstruction; the three parts are scientifically linked, since new transformations from (1) create new Bailey-type transforms and new embedded identities that face the same questions as (2) and (3), and an affirmative answer to (2) would systematically generate further identities in need of the interpretations sought in (3).

References

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