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Some More Identities of Kanade–Russell Type Derived Using Rosengren’s Method

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Abstract

In the present paper, we consider some variations and generalizations of the multi-sum to single-sum transformation recently used by Rosengren in his proof of the Kanade–Russell identities. These general transformations are then used to prove a number of identities equating multi-sums and infinite products or multi-sums and infinite product \(\times \) a false theta series. Examples include the following:

$$\begin{aligned}&\sum _{j,k,p,r=0}^{\infty } \frac{(-1)^{j+k} q^{ (2 j + k - p + r)^2/2 + k (k + 4)/2 + 3 j - p/2 + 3 r/2}(-q;q)_r}{\left( q^2;q^2\right) {}_j (q;q)_k (q;q)_p (q;q)_r} \\&\quad =2\frac{ \left( -q;q^2\right) {}_{\infty } \left( -q^2,-q^{14},q^{16};q^{16}\right) {}_{\infty }}{(q;q)_{\infty }}. \end{aligned}$$

Let

$$\begin{aligned} Q(i,j,k,l,p):= & {} \frac{1}{2} (i+6j+4 k+2 l-p) (i+6 j+4 k+2 l-p-1)\\&+2 k (k-1)+l (l-1)+3 i +15j+14k+5 l-2 p. \end{aligned}$$

Then

$$\begin{aligned}&\sum _{i,j,k,l,p=0}^{\infty } \frac{(-1)^{l+k}q^{Q(i,j,k,l,p)} }{ \left( q;q\right) {}_i \left( q^6;q^6\right) {}_j \left( q^4;q^4\right) {}_k \left( q^2;q^2\right) {}_l (q;q)_p } \\&\quad = \frac{2 (-q;q)_{\infty }^2 }{q \left( q^3;q^6\right) {}_{\infty } \left( q^4;q^4\right) {}_{\infty }}\left( 1+ \sum _{r=1}^{\infty } \left( q^{9 r^2+6 r}-q^{9 r^2-6 r}\right) \right) . \\&\sum _{j,k,p=0}^{\infty } (-1)^{k} \frac{q^{(3j+2k-p)(3j+2k-p-1)/2+k(k-1)-p+6j+6k} }{(q^3;q^3)_j(q^2;q^2)_{k} (q;q)_{p}}\\&\quad = \frac{ (-1;q)_{\infty }(q^{18};q^{18})_{\infty } }{ (q^3;q^3)_{\infty }(q^{9};q^{18})_{\infty } }. \end{aligned}$$

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Notes

  1. After substituting the right side of (3.17) for the left side in (3.16), and making the other indicated substitutions, the resulting identity is valid for \(|qd|<1\), by analytic continuation.

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  1. Mathematics Department, West Chester University, 25 University Avenue, West Chester, PA, 19383, USA

    James Mc Laughlin

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Communicated by Ken Ono.

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Mc Laughlin, J. Some More Identities of Kanade–Russell Type Derived Using Rosengren’s Method. Ann. Comb. 27, 329–352 (2023). https://doi.org/10.1007/s00026-022-00586-3

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