Gustafson-Rakha-Type Elliptic Hypergeometric Series

Hjalmar Rosengren

arXiv:1701.08960·math.CA·June 2, 2017

Gustafson-Rakha-Type Elliptic Hypergeometric Series

Hjalmar Rosengren

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TL;DR

This paper proves a new multivariable elliptic extension of Jackson's summation formula, generalizing previous results and introducing novel elliptic summations and Bailey transformations with broad mathematical implications.

Contribution

It provides the first multivariable elliptic extension of Jackson's summation conjecture, including new elliptic identities and transformations.

Findings

01

Proved a multivariable elliptic Jackson summation formula.

02

Derived two new multivariable elliptic Jackson summations.

03

Established two novel multivariable elliptic Bailey transformations.

Abstract

We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.

Equations112

\int \frac{1 \leq i < j \leq n \prod ( z _{i} / z _{j} ) _{\infty} ( z _{j} / z _{i} ) _{\infty} j = 1 \prod n ( S / z _{j} ) _{\infty}}{1 \leq i < j \leq n \prod ( t z _{i} z _{j} ) _{\infty} i = 1 \prod n ( j = 1 \prod 3 ( c _{j} z _{i} ) _{\infty} j = 1 \prod n ( d _{j} / z _{i} ) _{\infty} )} \frac{d z _{1}}{z _{1}} \dots \frac{d z _{n - 1}}{z _{n - 1}},

\int \frac{1 \leq i < j \leq n \prod ( z _{i} / z _{j} ) _{\infty} ( z _{j} / z _{i} ) _{\infty} j = 1 \prod n ( S / z _{j} ) _{\infty}}{1 \leq i < j \leq n \prod ( t z _{i} z _{j} ) _{\infty} i = 1 \prod n ( j = 1 \prod 3 ( c _{j} z _{i} ) _{\infty} j = 1 \prod n ( d _{j} / z _{i} ) _{\infty} )} \frac{d z _{1}}{z _{1}} \dots \frac{d z _{n - 1}}{z _{n - 1}},

∣ q ∣, ∣ t ∣, ∣ c_{1} ∣, ∣ c_{2} ∣, ∣ c_{3} ∣, ∣ d_{1} ∣, \dots, ∣ d_{n} ∣ < 1,

∣ q ∣, ∣ t ∣, ∣ c_{1} ∣, ∣ c_{2} ∣, ∣ c_{3} ∣, ∣ d_{1} ∣, \dots, ∣ d_{n} ∣ < 1,

\displaystyle\theta(z)=\prod\limits_{j=0}^{\infty}\big{(}1-p^{j}z\big{)}\big{(}1-p^{j+1}/z\big{)},

\displaystyle\theta(z)=\prod\limits_{j=0}^{\infty}\big{(}1-p^{j}z\big{)}\big{(}1-p^{j+1}/z\big{)},

\displaystyle(z)_{k}=\begin{cases}\theta(z)\theta(qz)\dotsm\theta\big{(}q^{k-1}z\big{)},&k\in\mathbb{Z}_{\geq 0},\\ 1/\theta\big{(}q^{k}z\big{)}\theta\big{(}q^{k+1}z\big{)}\dotsm\theta\big{(}q^{-1}z\big{)},&k\in\mathbb{Z}_{<0}\end{cases}

θ (z_{1}, \dots, z_{m}) = θ (z_{1}) \dots θ (z_{m}), (z_{1}, \dots, z_{m})_{k} = (z_{1})_{k} \dots (z_{m})_{k} .

θ (z_{1}, \dots, z_{m}) = θ (z_{1}) \dots θ (z_{m}), (z_{1}, \dots, z_{m})_{k} = (z_{1})_{k} \dots (z_{m})_{k} .

θ (1/ z) = θ (p z) = - θ (z) / z,

θ (1/ z) = θ (p z) = - θ (z) / z,

(a)_{n + k} = (a)_{n} (a q^{n})_{k}, (a)_{n - k} = (- 1)^{k} q^{(2 k)} (q^{1 - n} / a)^{k} \frac{( a ) _{n}}{( q ^{1 - n} / a ) _{k}},

\frac{Δ ( z q ^{x} )}{Δ ( z )} = 1 \leq i < j \leq n \prod \frac{q ^{x_{i}} θ ( q ^{x_{j} - x_{i}} z _{j} / z _{i} )}{θ ( z _{j} / z _{i} )},

\frac{Δ ( z q ^{x} )}{Δ ( z )} = 1 \leq i < j \leq n \prod \frac{q ^{x_{i}} θ ( q ^{x_{j} - x_{i}} z _{j} / z _{i} )}{θ ( z _{j} / z _{i} )},

\frac{Δ ( z q ^{x} )}{Δ ( z )} = (- 1)^{∣ x ∣} q^{- (2 ∣ x ∣) - ∣ x ∣} i, j = 1 \prod n \frac{( q z _{i} / z _{j} ) _{x_{i}}}{( q ^{- x_{j}} z _{i} / z _{j} ) _{x_{i}}}

\frac{Δ ( z q ^{x} )}{Δ ( z )} = (- 1)^{∣ x ∣} q^{- (2 ∣ x ∣) - ∣ x ∣} i, j = 1 \prod n \frac{( q z _{i} / z _{j} ) _{x_{i}}}{( q ^{- x_{j}} z _{i} / z _{j} ) _{x_{i}}}

k = 1 \sum n \frac{j = 1 \prod n + 1 θ ( z _{k} / b _{j} )}{θ ( z _{k} / t ) j = 1 , j \neq = k \prod n θ ( z _{k} / z _{j} )} = \frac{j = 1 \prod n + 1 θ ( b _{j} / t )}{j = 1 \prod n θ ( z _{j} / t )},

k = 1 \sum n \frac{j = 1 \prod n + 1 θ ( z _{k} / b _{j} )}{θ ( z _{k} / t ) j = 1 , j \neq = k \prod n θ ( z _{k} / z _{j} )} = \frac{j = 1 \prod n + 1 θ ( b _{j} / t )}{j = 1 \prod n θ ( z _{j} / t )},

\displaystyle\sum_{x=0}^{N}\frac{\theta\big{(}aq^{2x}\big{)}}{\theta(a)}\frac{\big{(}a,b,c,d,e,q^{-N}\big{)}_{x}q^{x}}{\big{(}q,aq/b,aq/c,aq/d,aq/e,aq^{N+1}\big{)}_{x}}=\frac{(aq,aq/bc,aq/bd,aq/cd)_{N}}{(aq/b,aq/c,aq/d,aq/bcd)_{N}},

\displaystyle\sum_{x=0}^{N}\frac{\theta\big{(}aq^{2x}\big{)}}{\theta(a)}\frac{\big{(}a,b,c,d,e,q^{-N}\big{)}_{x}q^{x}}{\big{(}q,aq/b,aq/c,aq/d,aq/e,aq^{N+1}\big{)}_{x}}=\frac{(aq,aq/bc,aq/bd,aq/cd)_{N}}{(aq/b,aq/c,aq/d,aq/bcd)_{N}},

\displaystyle\sum_{x=0}^{N}\frac{\theta\big{(}aq^{2x}\big{)}}{\theta(a)}\frac{\big{(}a,b,c,d,e,f,g,q^{-N}\big{)}_{x}q^{x}}{\big{(}q,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq^{N+1}\big{)}_{x}}

\displaystyle\sum_{x=0}^{N}\frac{\theta\big{(}aq^{2x}\big{)}}{\theta(a)}\frac{\big{(}a,b,c,d,e,f,g,q^{-N}\big{)}_{x}q^{x}}{\big{(}q,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq^{N+1}\big{)}_{x}}

\displaystyle=\frac{(aq,aq/ef,\lambda q/e,\lambda q/f)_{N}}{(\lambda q,\lambda q/ef,aq/e,aq/f)_{N}}\sum_{x=0}^{N}\frac{\theta\big{(}\lambda q^{2x}\big{)}}{\theta(\lambda)}\frac{\big{(}\lambda,\lambda b/a,\lambda c/a,\lambda d/a,e,f,g,q^{-N}\big{)}_{x}\,q^{x}}{\big{(}q,aq/b,aq/c,aq/d,\lambda q/e,\lambda q/f,\lambda q/g,\lambda q^{N+1}\big{)}_{x}}

\displaystyle\sum_{x_{1},\dots,x_{n}=0}^{N_{1},\dots,N_{n}}\frac{\Delta(zq^{x})}{\Delta(z)}\frac{\theta\big{(}aq^{2|x|}\big{)}}{\theta(a)}\frac{(a,b,c)_{|x|}\prod\limits_{i=1}^{n}(d/z_{i})_{|x|}}{\big{(}aq/b,aq/c,aq^{|N|+1}\big{)}_{|x|}\prod\limits_{i=1}^{n}\big{(}aq^{|N|+1-N_{i}}/ez_{i}\big{)}_{|x|}}q^{|x|}

\displaystyle\sum_{x_{1},\dots,x_{n}=0}^{N_{1},\dots,N_{n}}\frac{\Delta(zq^{x})}{\Delta(z)}\frac{\theta\big{(}aq^{2|x|}\big{)}}{\theta(a)}\frac{(a,b,c)_{|x|}\prod\limits_{i=1}^{n}(d/z_{i})_{|x|}}{\big{(}aq/b,aq/c,aq^{|N|+1}\big{)}_{|x|}\prod\limits_{i=1}^{n}\big{(}aq^{|N|+1-N_{i}}/ez_{i}\big{)}_{|x|}}q^{|x|}

\displaystyle\qquad\quad{}\times\prod\limits_{i=1}^{n}\frac{\big{(}aq^{|N|+1}/ez_{i}\big{)}_{|x|-x_{i}}(ez_{i})_{x_{i}}\prod\limits_{j=1}^{n}\big{(}q^{-N_{j}}z_{i}/z_{j}\big{)}_{x_{i}}}{(d/z_{i})_{|x|-x_{i}}(aqz_{i}/d)_{x_{i}}\prod\limits_{j=1}^{n}(qz_{i}/z_{j})_{x_{i}}}

= \frac{( a q , a q / b c ) _{∣ N ∣}}{( a q / b , a q / c ) _{∣ N ∣}} i = 1 \prod n \frac{( a q z _{i} / b d , a q z _{i} / c d ) _{N_{i}}}{( a q z _{i} / d , a q z _{i} / b c d ) _{N_{i}}} .

x_{1}, \dots, x_{n} \geq 0, x_{1} + \dots + x_{n} = N \sum \frac{Δ ( z q ^{x} )}{Δ ( z )} 1 \leq i < j \leq n \prod q^{x_{i} x_{j}} (z_{i} z_{j})_{x_{i} + x_{j}} i = 1 \prod n \frac{j = 1 \prod 4 ( z _{i} b _{j} ) _{x_{i}}}{z _{i}^{x_{i}} j = 1 \prod n ( q z _{i} / z _{j} ) _{x_{i}}}

x_{1}, \dots, x_{n} \geq 0, x_{1} + \dots + x_{n} = N \sum \frac{Δ ( z q ^{x} )}{Δ ( z )} 1 \leq i < j \leq n \prod q^{x_{i} x_{j}} (z_{i} z_{j})_{x_{i} + x_{j}} i = 1 \prod n \frac{j = 1 \prod 4 ( z _{i} b _{j} ) _{x_{i}}}{z _{i}^{x_{i}} j = 1 \prod n ( q z _{i} / z _{j} ) _{x_{i}}}

= ⎩ ⎨ ⎧ \frac{( Z b _{1} , Z b _{2} , Z b _{3} , Z b _{4} ) _{N}}{Z ^{N} ( q ) _{N}}, \frac{( Z , Z b _{1} b _{2} , Z b _{1} b _{3} , Z b _{1} b _{4} ) _{N}}{( Z b _{1} ) ^{N} ( q ) _{N}}, n odd, n even .

k = 1 \sum n \frac{j = 1 \prod 4 θ ( z _{k} b _{j} )}{z _{k}} j = 1, j \neq = k \prod n \frac{θ ( z _{k} z _{j} )}{θ ( z _{k} / z _{j} )} = {θ (Z b_{1}, Z b_{2}, Z b_{3}, Z b_{4}) / Z, θ (Z, Z b_{1} b_{2}, Z b_{1} b_{3}, Z b_{1} b_{4}) / Z b_{1}, n odd, n even .

k = 1 \sum n \frac{j = 1 \prod 4 θ ( z _{k} b _{j} )}{z _{k}} j = 1, j \neq = k \prod n \frac{θ ( z _{k} z _{j} )}{θ ( z _{k} / z _{j} )} = {θ (Z b_{1}, Z b_{2}, Z b_{3}, Z b_{4}) / Z, θ (Z, Z b_{1} b_{2}, Z b_{1} b_{3}, Z b_{1} b_{4}) / Z b_{1}, n odd, n even .

f (w) = f (b) \frac{θ ( c w , c / w )}{θ ( c b , c / b )} + f (c) \frac{θ ( b w , b / w )}{θ ( b c , b / c )},

f (w) = f (b) \frac{θ ( c w , c / w )}{θ ( c b , c / b )} + f (c) \frac{θ ( b w , b / w )}{θ ( b c , b / c )},

\displaystyle\frac{1}{2}\left({(-1)^{n+1}Z\prod\limits_{j=1}^{4}\theta(b_{j})}+{Z\prod\limits_{j=1}^{4}\theta(-b_{j})}+\frac{1}{\sqrt{p}}{\prod\limits_{j=1}^{4}\theta\big{(}\sqrt{p}b_{j}\big{)}}-\frac{1}{\sqrt{p}}{\prod\limits_{j=1}^{4}\theta\big{(}{-}\sqrt{p}b_{j}\big{)}}\right).

\displaystyle\frac{1}{2}\left({(-1)^{n+1}Z\prod\limits_{j=1}^{4}\theta(b_{j})}+{Z\prod\limits_{j=1}^{4}\theta(-b_{j})}+\frac{1}{\sqrt{p}}{\prod\limits_{j=1}^{4}\theta\big{(}\sqrt{p}b_{j}\big{)}}-\frac{1}{\sqrt{p}}{\prod\limits_{j=1}^{4}\theta\big{(}{-}\sqrt{p}b_{j}\big{)}}\right).

\displaystyle R_{N+1}(Z;b_{1},b_{2},b_{3},b_{4})=\frac{q^{N}\theta(q)}{\theta(q^{N+1})}R_{1}\big{(}q^{N}Z;q^{-N}b_{1},b_{2},b_{3},b_{4}\big{)}R_{N}(Z;qb_{1},b_{2},b_{3},b_{4}).

\displaystyle R_{N+1}(Z;b_{1},b_{2},b_{3},b_{4})=\frac{q^{N}\theta(q)}{\theta(q^{N+1})}R_{1}\big{(}q^{N}Z;q^{-N}b_{1},b_{2},b_{3},b_{4}\big{)}R_{N}(Z;qb_{1},b_{2},b_{3},b_{4}).

\displaystyle R_{N+1}(Z;b_{1},b_{2},b_{3},b_{4})=\frac{q^{N}\theta(q)}{\theta(q^{N+1})}R_{1}\big{(}q^{N}Z;q^{-N}b_{1},b_{2},b_{3},b_{4}\big{)}

\displaystyle R_{N+1}(Z;b_{1},b_{2},b_{3},b_{4})=\frac{q^{N}\theta(q)}{\theta(q^{N+1})}R_{1}\big{(}q^{N}Z;q^{-N}b_{1},b_{2},b_{3},b_{4}\big{)}

\times x_{1}, \dots, x_{n} \geq 0, x_{1} + \dots + x_{n} = N \sum \frac{Δ ( z q ^{x} )}{Δ ( z )} 1 \leq i < j \leq n \prod q^{x_{i} x_{j}} (z_{i} z_{j})_{x_{i} + x_{j}} i = 1 \prod n \frac{( q z _{i} b _{1} ) _{x_{i}} j = 2 \prod 4 ( z _{i} b _{j} ) _{x_{i}}}{z _{i}^{x_{i}} j = 1 \prod n ( q z _{i} / z _{j} ) _{x_{i}}},

R_{N + 1} (Z; b_{1}, b_{2}, b_{3}, b_{4})

R_{N + 1} (Z; b_{1}, b_{2}, b_{3}, b_{4})

= \frac{q ^{N} θ ( q )}{θ ( q ^{N + 1} )} x_{1}, \dots, x_{n} \geq 0, x_{1} + \dots + x_{n} = N \sum y_{1}, \dots, y_{n} \geq 0, y_{1} + \dots + y_{n} = 1 \sum \frac{Δ ( z q ^{x + y} )}{Δ ( z )} 1 \leq i < j \leq n \prod q^{x_{i} x_{j}} (z_{i} z_{j})_{x_{i} + x_{j} + y_{i} + y_{j}}

\displaystyle\qquad\quad{}\times\prod\limits_{i=1}^{n}\frac{(qz_{i}b_{1})_{x_{i}}\big{(}q^{x_{i}-N}z_{i}b_{1}\big{)}_{y_{i}}\prod\limits_{j=2}^{4}(z_{i}b_{j})_{x_{i}+y_{i}}}{z_{i}^{x_{i}}(z_{i}q^{x_{i}})^{y_{i}}\prod\limits_{j=1}^{n}(qz_{i}/z_{j})_{x_{i}}\big{(}q^{1+x_{i}-x_{j}}z_{i}/z_{j}\big{)}_{y_{i}}}

= \frac{q ^{N} θ ( q )}{θ ( q ^{N + 1} )} x_{1}, \dots, x_{n} \geq 0, x_{1} + \dots + x_{n} = N + 1 \sum y_{1}, \dots, y_{n} \geq 0, y_{1} + \dots + y_{n} = 1 \sum \frac{Δ ( z q ^{x} )}{Δ ( z )} 1 \leq i < j \leq n \prod q^{x_{i} x_{j} - x_{i} y_{j} - x_{j} y_{i}} (z_{i} z_{j})_{x_{i} + x_{j}}

\displaystyle\qquad\quad{}\times\prod\limits_{i=1}^{n}\frac{(qz_{i}b_{1})_{x_{i}-y_{i}}\big{(}q^{x_{i}-y_{i}-N}z_{i}b_{1}\big{)}_{y_{i}}\prod\limits_{j=2}^{4}(z_{i}b_{j})_{x_{i}}}{z_{i}^{x_{i}}q^{(x_{i}-y_{i})y_{i}}\prod\limits_{j=1}^{n}(qz_{i}/z_{j})_{x_{i}-y_{i}}\big{(}q^{1+x_{i}-x_{j}-y_{i}+y_{j}}z_{i}/z_{j}\big{)}_{y_{i}}},

i < j \prod q^{x_{i} y_{j} + x_{j} y_{i}} i \prod q^{(x_{i} - y_{i}) y_{i}} = q^{(x_{1} + \dots + x_{n}) (y_{1} + \dots + y_{n}) - (y_{1}^{2} + \dots + y_{n}^{2})} = q^{(N + 1) \cdot 1 - 1} = q^{N},

i < j \prod q^{x_{i} y_{j} + x_{j} y_{i}} i \prod q^{(x_{i} - y_{i}) y_{i}} = q^{(x_{1} + \dots + x_{n}) (y_{1} + \dots + y_{n}) - (y_{1}^{2} + \dots + y_{n}^{2})} = q^{(N + 1) \cdot 1 - 1} = q^{N},

R_{N + 1} (Z; b_{1}, b_{2}, b_{3}, b_{4})

R_{N + 1} (Z; b_{1}, b_{2}, b_{3}, b_{4})

= \frac{1}{θ ( q ^{N + 1} )} x_{1}, \dots, x_{n} \geq 0, x_{1} + \dots + x_{n} = N + 1 \sum \frac{Δ ( z q ^{x} )}{Δ ( z )} 1 \leq i < j \leq n \prod q^{x_{i} x_{j}} (z_{i} z_{j})_{x_{i} + x_{j}} i = 1 \prod n \frac{( q z _{i} b _{1} ) _{x_{i}} j = 2 \prod 4 ( z _{i} b _{j} ) _{x_{i}}}{z _{i}^{x_{i}} j = 1 \prod n ( q z _{i} / z _{j} ) _{x_{i}}}

\displaystyle\qquad{}\times\sum_{\begin{subarray}{c}y_{1},\dots,y_{n}\geq 0,\\ y_{1}+\dots+y_{n}=1\end{subarray}}\prod\limits_{i=1}^{n}\frac{\big{(}q^{x_{i}-y_{i}-N}z_{i}b_{1}\big{)}_{y_{i}}\prod\limits_{j=1}^{n}\big{(}q^{1+x_{i}-y_{i}}z_{i}/z_{j}\big{)}_{y_{i}}}{\big{(}q^{1+x_{i}-y_{i}}z_{i}b_{1}\big{)}_{y_{i}}\prod\limits_{j=1,\,j\neq i}^{n}\big{(}q^{x_{i}-x_{j}}z_{i}/z_{j}\big{)}_{y_{i}}}.

\displaystyle\sum_{k=1}^{n}\frac{\theta\big{(}q^{x_{k}-N-1}z_{k}b_{1}\big{)}\prod\limits_{j=1}^{n}\theta(q^{x_{k}}z_{k}/z_{j})}{\theta(q^{x_{k}}z_{k}b_{1})\prod\limits_{j=1,\,j\neq k}^{n}\theta(q^{x_{k}-x_{j}}z_{k}/z_{j})}.

\displaystyle\sum_{k=1}^{n}\frac{\theta\big{(}q^{x_{k}-N-1}z_{k}b_{1}\big{)}\prod\limits_{j=1}^{n}\theta(q^{x_{k}}z_{k}/z_{j})}{\theta(q^{x_{k}}z_{k}b_{1})\prod\limits_{j=1,\,j\neq k}^{n}\theta(q^{x_{k}-x_{j}}z_{k}/z_{j})}.

\displaystyle\theta\big{(}q^{N+1}\big{)}\prod\limits_{i=1}^{n}\frac{\theta(z_{i}b_{1})}{\theta(q^{x_{i}}z_{i}b_{1})}=\theta\big{(}q^{N+1}\big{)}\prod\limits_{i=1}^{n}\frac{(z_{i}b_{1})_{x_{i}}}{(qz_{i}b_{1})_{x_{i}}}

\displaystyle\theta\big{(}q^{N+1}\big{)}\prod\limits_{i=1}^{n}\frac{\theta(z_{i}b_{1})}{\theta(q^{x_{i}}z_{i}b_{1})}=\theta\big{(}q^{N+1}\big{)}\prod\limits_{i=1}^{n}\frac{(z_{i}b_{1})_{x_{i}}}{(qz_{i}b_{1})_{x_{i}}}

\displaystyle\sum_{\begin{subarray}{c}x_{1},\dots,x_{n}\geq 0,\\ x_{1}+\dots+x_{n}\leq N\end{subarray}}\frac{\Delta(zq^{x})}{\Delta(z)}\prod\limits_{i=1}^{n}\frac{\theta(az_{i}q^{|x|+x_{i}})}{\theta(az_{i})}\frac{\prod\limits_{1\leq i<j\leq n}(z_{i}z_{j})_{x_{i}+x_{j}}}{\prod\limits_{i=1}^{n}(aq/z_{i})_{|x|-x_{i}}}\frac{\big{(}q^{-N}\big{)}_{|x|}\prod\limits_{i=1}^{n}(az_{i})_{|x|}}{\prod\limits_{j=1}^{4}(aq/b_{j})_{|x|}}q^{|x|}

\displaystyle\sum_{\begin{subarray}{c}x_{1},\dots,x_{n}\geq 0,\\ x_{1}+\dots+x_{n}\leq N\end{subarray}}\frac{\Delta(zq^{x})}{\Delta(z)}\prod\limits_{i=1}^{n}\frac{\theta(az_{i}q^{|x|+x_{i}})}{\theta(az_{i})}\frac{\prod\limits_{1\leq i<j\leq n}(z_{i}z_{j})_{x_{i}+x_{j}}}{\prod\limits_{i=1}^{n}(aq/z_{i})_{|x|-x_{i}}}\frac{\big{(}q^{-N}\big{)}_{|x|}\prod\limits_{i=1}^{n}(az_{i})_{|x|}}{\prod\limits_{j=1}^{4}(aq/b_{j})_{|x|}}q^{|x|}

\displaystyle\qquad\quad{}\times\prod\limits_{i=1}^{n}\frac{\prod\limits_{j=1}^{4}(z_{i}b_{j})_{x_{i}}}{\big{(}aq^{N+1}z_{i}\big{)}_{x_{i}}\prod\limits_{j=1}^{n}(qz_{i}/z_{j})_{x_{i}}}

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Full text

\FirstPageHeading

\ShortArticleName

Gustafson–Rakha-Type Elliptic Hypergeometric Series

\ArticleName

Gustafson–Rakha-Type Elliptic Hypergeometric

Series††This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications. The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.html

\Author

Hjalmar ROSENGREN

\AuthorNameForHeading

H. Rosengren

\Address

Department of Mathematical Sciences, Chalmers University of Technology and

University of Gothenburg, SE-412 96 Gothenburg, Sweden \Email[email protected] \URLaddresshttp://www.math.chalmers.se/~hjalmar/

\ArticleDates

Received February 02, 2017, in final form May 29, 2017; Published online June 01, 2017

\Abstract

We prove a multivariable elliptic extension of Jackson’s summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.

\Keywords

elliptic hypergeometric series; multivariable hypergeometric series; Jackson summation; Bailey transformation

\Classification

33D67

1 Introduction

By combining two integral evaluations previously obtained by Gustafson [6], Gustafson and Rakha [7] evaluated the basic hypergeometric integral

[TABLE]

where the integration is over $|z_{1}|=\dots=|z_{n-1}|=1$ , $(z)_{\infty}=\prod\limits_{j=0}^{\infty}(1-q^{j}z)$ , the parameters satisfy

[TABLE]

$z_{n}$ is determined from the integration variables through $z_{1}\dotsm z_{n}=1$ and $S=t^{n-2}c_{1}c_{2}c_{3}d_{1}\dotsm d_{n}$ . By applying residue calculus to (1.1), they could evaluate a certain multivariable basic hypergeometric finite sum, equivalent to the case $p=0$ of Theorem 3.1 below.

Since the seminal work of Date et al. [4] and Frenkel and Turaev [5], it has been recognized that basic hypergeometric functions appear as the trigonometric limit of more general elliptic hypergeometric functions. Elliptic extensions of Gustafson’s two integral evaluations mentioned above were conjectured in [16, 19] and proved in [10]. Spiridonov [16] used these (at the time conjectural) evaluations to obtain an elliptic extension of (1.1). He also stated the corresponding summation formula as a conjecture. Although it seems likely that this conjecture can be deduced from Spiridonov’s integral, such a derivation is still missing from the literature. The purpose of the present paper is to give a direct proof of Spiridonov’s conjectured summation and to apply it to derive some further summation and transformation formulas.

It is worth mentioning that Spiridonov’s elliptic extension of (1.1) can be interpreted as the identity between superconformal indices of two dual quantum field theories [17, Sections 12.1.2–12.1.3]. This indicates that (1.1) and related results are not mere curiosities and that it is not unreasonable to expect further applications.

The plan of the paper is as follows. In Section 3, we prove Spiridonov’s conjecture. The proof is elementary and provides in particular a significant simplification of the trigonometric case. The only previously known proof of the Gustafson–Rakha summation is the original one, which as we recall is based on first proving two auxiliary multiple integral evaluations, combining them to obtain (1.1) and finally on a technical computation to pass from integrals to finite residue sums. In Section 4, we give some applications of our result. Namely, combining the elliptic Gustafson–Rakha sum with a summation from [14], we obtain two transformation formulas and two further summation formulas for multivariable elliptic hypergeometric series. These four results are all new even in the trigonometric case.

Note added in proof: After completing this work, I learned from Masahiko Ito and Masatoshi Noumi that they have independently proved Theorem 3.1, using a different method.

2 Preliminaries

When $z=(z_{1},\dots,z_{n})$ is a vector we will write $|z|=z_{1}+\dots+z_{n}$ and $Z=z_{1}\dotsm z_{n}$ .

Throughout, $p$ and $q$ will be fixed parameters with $|p|<1$ . We employ the standard notation

[TABLE]

as well as

[TABLE]

Most of our computations are based on the elementary identities

[TABLE]

which will be used without comment. All our sums contain the $A$ -type factor

[TABLE]

where $z=(z_{1},\dots,z_{n})$ and $x=(x_{1},\dots,x_{n})$ is the summation index. We mention the useful identity [12, equation (3.8)]

[TABLE]

and [21, Example 20.53.3]

[TABLE]

valid for $tz_{1}\dotsm z_{n}=b_{1}\dotsm b_{n+1}$ .

In the one-variable case, the most fundamental results for elliptic hypergeometric series are the elliptic Jackson (or Frenkel–Turaev) summation

[TABLE]

valid for $a^{2}q^{N+1}=bcde$ , and the elliptic Bailey transformation

[TABLE]

valid for $a^{3}q^{N+2}=bcdefg$ , $\lambda=a^{2}q/bcd$ [5]. Numerous multivariable extensions of (2.3) and (2.4) are known, see, e.g., [3, 9, 11, 12, 13, 14, 18, 20]; further examples are obtained in the present paper. We will need one such result, a multivariable extension of (2.3) obtained in [14] (see [15] for the case $p=0$ ). Namely, for $a^{2}q^{|N|+1}=bcde$ ,

[TABLE]

3 The elliptic Gustafson–Rakha summation

Our main result is the following identity. It is easy to see that the case $p=0$ is equivalent to [7, Theorem 1.2] and the general case to the conjecture of [16, p. 953]. Recall the notation $Z=z_{1}\dotsm z_{n}$ .

Theorem 3.1.

For parameters subject to $q^{N-1}b_{1}\dotsm b_{4}z_{1}^{2}\dotsm z_{n}^{2}=1$ ,

[TABLE]

We will prove Theorem 3.1 by induction on $N$ . In the case $N=1$ , we have $x_{i}=\delta_{ik}$ for some $k$ . Using $k$ as summation index, Theorem 3.1 reduces to the following theta function identity.

Lemma 3.2.

For parameters subject to $b_{1}b_{2}b_{3}b_{4}z_{1}^{2}\dotsm z_{n}^{2}=1$ ,

[TABLE]

Proof.

We apply induction on $n$ , starting from the trivial case $n=1$ . Let $b_{1}=vw$ and $b_{2}=v/w$ , with $v$ and $w$ free parameters. As a function of $w$ , each term in the sum, as well as the right-hand side, has the form $f(w)=C\theta(aw,a/w)$ with $C$ and $a$ independent of $w$ . It is a classical fact that any such function is determined by its values at two generic points. Indeed, Weierstrass’ identity (which is equivalent to the case $n=2$ of (2.2)) states that

[TABLE]

provided that $bc,b/c\notin p^{\mathbb{Z}}$ . Thus, it suffices to verify (3.2) for two independent values of $b_{1}$ . Assuming $n\geq 2$ , we choose $b_{1}=1/z_{n-1}$ and $b_{1}=1/z_{n}$ . By symmetry, it is enough to consider the second case. Then, the term corresponding to $k=n$ cancels and we are reduced to an identity equivalent to (3.2), with $n$ replaced by $n-1$ and $b_{1}$ by $z_{n}$ . ∎

We mention that it is not hard to deduce (3.2) from classical theta function identities. Indeed, let $t=b_{n+1}$ in (2.2) (so that the right-hand side vanishes) and then make the substitutions $n\mapsto n+4$ , $(z_{1},\dots,z_{n})\mapsto(z_{1},\dots,z_{n},1,-1,\sqrt{p},-1/\sqrt{p})$ , $(b_{1},\dots,b_{n})\mapsto(z_{1}^{-1},\dots,z_{n}^{-1},b_{1}^{-1},b_{2}^{-1}$ , $b_{3}^{-1},b_{4}^{-1})$ . Using the elementary identities $\theta(z^{2})=\theta(z,-z,\sqrt{p}z,-\sqrt{p}z)$ and $2=\theta(-1,\sqrt{p},-\sqrt{p})$ , one may deduce that the left-hand side of (3.2) is equal to

[TABLE]

The fact that this equals the right-hand side of (3.2) follows from Jacobi’s fundamental formulae [21, Section 21.22].

The inductive step in the proof of Theorem 3.1 is almost identical to that of [12, Theorem 5.1]. Denoting the right-hand side of (3.1) by $R_{N}(Z;b_{1},b_{2},b_{3},b_{4})$ (where we for a moment consider $Z$ as a free variable) we observe that, regardless of the parity of $n$ ,

[TABLE]

Assuming (3.1) for fixed $N$ , it follows that

[TABLE]

where $Z=z_{1}\dotsm z_{n}$ and $q^{N}BZ^{2}=1$ . We pull the factor $R_{1}$ inside the sum and expand it using (3.1), with $z_{i}$ replaced by $q^{x_{i}}z_{i}$ . This gives

[TABLE]

where we replaced each $x_{i}$ by $x_{i}-y_{i}$ and used that $y_{i}y_{j}=0$ for $i\neq j$ .

By elementary manipulations, using

[TABLE]

the expression above can be rewritten

[TABLE]

Writing $y_{i}=\delta_{ik}$ , the inner sum takes the form

[TABLE]

By (2.2), this can be evaluated as

[TABLE]

and we arrive at (3.1) with $N$ replaced by $N+1$ . This completes the proof of Theorem 3.1.

We will now rewrite (3.1) in a way that hides some of its symmetry but makes it clear that it generalizes the Frenkel–Turaev summation (2.3). To this end, we replace $n$ by $n+1$ , $z_{n+1}$ by $q^{-N}a^{-1}$ and eliminate $x_{n+1}$ from the summation. After routine simplification, we arrive at the following identity.

Corollary 3.3.

Assuming $a^{2}q^{N+1}=b_{1}b_{2}b_{3}b_{4}z_{1}^{2}\dotsm z_{n}^{2}$ ,

[TABLE]

4 Applications

The elliptic Bailey transformation (2.4) can be derived from the elliptic Jackson summation (2.3). Similar arguments can be used in multivariable situations, see, e.g., [1, 2, 8] for the trigonometric and [12] for the elliptic case. We will use this method to derive a new multivariable elliptic Bailey transformation by combining the two multivariable elliptic Jackson summations (2.5) and (3.3).

Theorem 4.1.

Suppose that $a^{3}q^{N+2}=bcdefgz_{1}^{2}\dotsm z_{n}^{2}$ and let $\lambda=a^{2}q/bcd$ . Then,

[TABLE]

Proof.

If we substitute

[TABLE]

in (2.5), the right-hand side takes the form

[TABLE]

Thus, the left-hand side of (4.1) can be expressed as

[TABLE]

We change the order of summation and replace the vector $x$ by $x+y$ . Some elementary manipulation, using in particular (2.1), gives

[TABLE]

We observe that the inner sum is as in Corollary 3.3, with the substitutions

[TABLE]

When $n$ is odd, the value of this sum can be rewritten

[TABLE]

which leads to the right-hand side of (4.1). The case of even $n$ is treated similarly. ∎

One may obtain further transformation formulas by iterating Theorem 4.1. We will only give one example, exploiting the fact that the left-hand side of (4.1) is invariant under interchanging $c$ and $e$ . In the identity expressing the corresponding symmetry of the right-hand side, we make the substitutions $(\lambda,a,b,c,d)\mapsto(a,a^{2}q/bcd,aq/cd,aq/bd,aq/bc)$ , keeping $e,f,g,z_{1},\dots,z_{n}$ fixed. This leads to another multivariable elliptic Bailey transformation.

Corollary 4.2.

Suppose that $a^{3}q^{N+2}=bcdefgz_{1}^{2}\dotsm z_{n}^{2}$ and let $\lambda=a^{2}q/bde$ . Then,

[TABLE]

Theorem 4.1 reduces to Corollary 3.3 when $aq=bc$ . More interestingly, when $b=1$ the left-hand side of (4.1) reduces to $1$ . After a change of parameters, this leads to the following new multivariable elliptic Jackson summation.

Corollary 4.3.

If $a^{2}q^{N+1}=bcdez_{1}^{2}\dotsm z_{n}^{2}$ , then

[TABLE]

Examining the proof of Theorem 4.1, we see that Corollary 4.3 is obtained by combining (3.3) with the special case $aq=bc$ of (2.5), when the right-hand side is equal to $\prod\limits_{i=1}^{n}\delta_{N_{i},0}$ . The latter identity can be viewed as a matrix inversion [14], so Corollary 4.3 is an inverted version of Corollary 3.3, just as (2.5) is an inverted version of the standard $A$ -type summation [12, Corollary 5.2].

If we let $\lambda d/a=1$ in Corollary 4.2, we obtain yet another multivariable elliptic Jackson summation. After a change of parameters, it takes the form

[TABLE]

valid for $a^{2}q^{N+1}=bcdez_{1}^{2}\dotsm z_{n}^{2}$ and $t$ arbitrary. This identity is less novel than Corollary 4.3, as it can be deduced from Theorem 3.1 in a more direct manner. Indeed, writing the sum as

[TABLE]

the inner sum is computed by Theorem 3.1 and the outer sum by (2.3). In fact, the same proof gives the following more general result, which reduces to (4.2) when $(d,e)=(fZ,gZ)$ or $(Z,fgZ)$ if $n$ is odd or even, respectively.

Corollary 4.4.

For parameters subject to $a^{2}q^{N+1}=bcde$ , $fghz_{1}^{2}\dotsm z_{n}^{2}=t$ ,

[TABLE]

Acknowledgements

This research is supported by the Swedish Science Research Council (Vetenskapsrådet). I would like to thank the anonymous referee for a very careful reading of the manuscript, leading to many improvements.

Bibliography21

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

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TL;DR

Contribution

Findings

Abstract

Peer Reviews

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1 Introduction

2 Preliminaries

3 The elliptic Gustafson–Rakha summation

Theorem 3.1**.**

Lemma 3.2**.**

Proof.

Corollary 3.3**.**

4 Applications

Theorem 4.1**.**

Proof.

Corollary 4.2**.**

Corollary 4.3**.**

Corollary 4.4**.**

Acknowledgements

Theorem 3.1.

Lemma 3.2.

Corollary 3.3.

Theorem 4.1.

Corollary 4.2.

Corollary 4.3.

Corollary 4.4.