-----------message from Christian Krathenthaler about a classical derivation -----------of Lemma of the paper kirillov.tex------------------------- From KRATT@Pap.UniVie.AC.AT Wed Nov 29 09:26:25 1995 Received: from Pap.UniVie.AC.AT (apap4.pap.univie.ac.at [131.130.11.14]) by euclid.math.temple.edu (8.6.12/8.6.12) with ESMTP id JAA26569 for ; Wed, 29 Nov 1995 09:26:17 -0500 From: KRATT@Pap.UniVie.AC.AT Posted-Date: Wed, 29 Nov 1995 09:26:17 -0500 Received-Date: Wed, 29 Nov 1995 09:26:17 -0500 Received: from Pap.UniVie.AC.AT by Pap.UniVie.AC.AT (PMDF V5.0-4 #10670) id <01HY7XTH9AK0AIBPLE@Pap.UniVie.AC.AT> for zeilberg@euclid.math.temple.edu; Wed, 29 Nov 1995 15:25:55 +0100 (MET) Date: Wed, 29 Nov 1995 15:25:55 +0100 (MET) Subject: the irrelevant identity To: zeilberg@euclid.math.temple.edu Message-id: <01HY7XTH9K76AIBPLE@Pap.UniVie.AC.AT> X-VMS-To: IN%ZEILBERGER X-VMS-Cc: KRATT MIME-version: 1.0 Content-transfer-encoding: 7BIT Status: RO Dear Doron, Thanks a lot for your nice "shorty". I know that it is "irrelevant". But maybe you are still interested to know a "classical" proof of your identity in Lemma 2. It is a special case of one of the "strange evaluations", namely of the identity (this is Plain-\TeX\ compatible) $$ \displaylines{ \sum_{j = 0}^{\infty}{q^j} {{(1-{\let \over / a {q^{4 j}}})} \over {({\let \over / a})}} {{({\let \over / e, {b\over e}}; q) _{j}} \over {({\let \over / {{a {q^3}}\over e}, {{a e {q^3}}\over b}}; {q^3}) _{j}}} {{({\let \over / {{a q}\over b}, {{a {q^2}}\over b}}; {q^2}) _{j}} \over {({\let \over / b q, b}; {q^2}) _{j}}} {{({\let \over / a, b}; {q^3}) _{j}} \over {({\let \over / q, {{a q}\over b}}; q) _{j}}} \cr = { {(\let \over / {b\over a}, {q\over b}, {{{q^2}}\over b}, {e\over a}, {{a {q^3}}\over b}, e q, e {q^2}, {{a e {q^3}}\over {{b^2}}} ;{q^3}) _\infty} \over {(\let \over / {1\over a}, q, {q^2}, {{b e}\over a}, {{a {q^3}}\over {{b^2}}}, {{e q}\over b}, {{e {q^2}}\over b}, {{a e {q^3}}\over b} ;{q^3}) _\infty} }, } $$ whenever the series terminates (this is somewhere in my unpublished paper "Some quadratic, cubic, quartic very well-poised summation formulas for basic hypergeometric series", one of the reasons that it is unpublished being that, for instance, the above identity follows from (3.8.19) in Gasper and Rahman's book by doing the replacements $a\to e$, $b\to b/e$, $d\to a$, $c\to a/e$, in this order, then only the first term on the left-hand side of (3.8.19) survives if we assume that $aq/b$ is of the form $q^-N$ with $N$ a nonnegative integer, and the surviving term is an $_8\phi_7$-sum which can be evaluated by means of Jackson's sum). In this identity do the replacements $b\to aq^{1+m}$, $a\to a^2$, $e\to a$, to obtain $$ \displaylines{ \sum_{j = 0}^{\infty}{q^j} {{({\let \over / {a^2} {q^{4 j}}})} \over {({\let \over / {a^2}})}} {{({\let \over / {a^2}, {a^2} {q^{1 + m}}}; {q^3}) _{j}} \over {({\let \over / q, {q^{-m}}}; q) _{j}}} {{({\let \over / {q^{-m}}, {q^{1 - m}}}; {q^2}) _{j}} \over {({\let \over / {a^2} {q^{2 + m}}, {a^2} {q^{1 + m}}}; {q^2}) _{j}}} {{({\let \over / a, a {q^{1 + m}}}; q) _{j}} \over {({\let \over / a {q^3}, a {q^{2 - m}}}; {q^3}) _{j}}} \cr = { {(\let \over / {q^{1 + m}}, {1\over {{a^2} {q^m}}}, {{{q^{1 - m}}}\over {{a^2}}}, {1\over a}, {q^{2 - m}}, a q, a {q^2}, {{{q^{1 - 2 m}}}\over a} ;{q^3}) _\infty} \over {(\let \over / {a^{-2}}, q, {q^2}, a {q^{1 + m}}, {{{q^{1 - 2 m}}}\over {{a^2}}}, {1\over {a {q^m}}}, {{{q^{1 - m}}}\over a}, a {q^{2 - m}} ;{q^3}) _\infty} }, } $$ and then let $a\to\infty$. There are some manipulations to do on the right-hand side, depending on whether $m$ is $\equiv 0$ or $\equiv 1$ mod 3. In case $m\equiv 2$ mod 3 it is obvious from the term $(q^{2-m};q^3)_\infty$ that the right-hand side vanishes. All the best, Christian