Analytic continuation of $_4F_3(1)$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:27:50Z http://mathoverflow.net/feeds/question/51123 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51123/analytic-continuation-of-4f-31 Analytic continuation of $_4F_3(1)$ Dmitri 2011-01-04T14:24:31Z 2011-01-05T04:40:45Z <p>The Gauss theorem $${_2F_1}(a,b;c;1)=\frac{\Gamma(c-a)\Gamma(c-b)}{\Gamma(c)\Gamma(c-a-b)}$$ allows to compute the analytic continuation of ${_2F_1}(a,b;c;1)$ for $a+b>c$ when the series definition diverges. The same can be done for $_3F_2(1)$ via Thomae relations. My question is how to find analytic continuation for $_4F_3(1)$ to the subdomains of $\mathbb{C}^7$ where the sum of the upper parameters is greater than the sum of the lower parameters, so that the series diverges. I know that no direct analogues of Thomae relations exit in this case so the formulas may be more complicated. I have seen some work by Allen Miller and other authors giving transformations for $_4F_3(1)$, but these transformations leave the excess (total upper parameters minus total lower parameters) invariant, so that divergent series is transformed into divergent series. The same question, of course, pertains to <code>$_pF_{p-1}(1)$</code> with $p>4$...</p> <p>Any help is highly appreciated.</p> http://mathoverflow.net/questions/51123/analytic-continuation-of-4f-31/51159#51159 Answer by Ira Gessel for Analytic continuation of $_4F_3(1)$ Ira Gessel 2011-01-04T20:50:40Z 2011-01-04T20:50:40Z <p>The documentation to Christian Krattenthaler's HYP package contains many contiguous relations for arbitrary $_pF_q$s. In particular, C20 on page 17 of <a href="http://www.mat.univie.ac.at/~kratt/hyp_hypq/hypm.pdf" rel="nofollow">http://www.mat.univie.ac.at/~kratt/hyp_hypq/hypm.pdf</a> looks as though it might help.</p> <p>The URL for the whole HYP package is <a href="http://www.mat.univie.ac.at/~kratt/hyp_hypq/hyp.html" rel="nofollow">http://www.mat.univie.ac.at/~kratt/hyp_hypq/hyp.html</a>.</p> http://mathoverflow.net/questions/51123/analytic-continuation-of-4f-31/51175#51175 Answer by Wadim Zudilin for Analytic continuation of $_4F_3(1)$ Wadim Zudilin 2011-01-05T02:49:00Z 2011-01-05T04:40:45Z <p>Dmitri, I am confused by your way to continue analytically to a point rather than to a domain. First of all, the Gauss summation formula is valid only if $\operatorname{Re}(c-a-b)>0$, so that your value at 1 is only a formal quantity assigned to the right-hand side (the ratio of gamma functions) when $a+b>c$. "The same can be done for ${}_3F_2$ via Thomae's transformations" is not correct by the same reason. So, your question is not about analytic continuation: you cannot continue the Gauss series at 1, viewed as an analytic function of three variables $a$, $b$ and $c$, through the barrier $\operatorname{Re}(c-a-b)=0$.</p> <p>If you look for a former way of assigning some reasonable values to the hypergeometric functions ${}_p F _{\substack{p-1}}$ at 1, it is naturally to play with the classical integral representations: the $(p-1)$-fold integral due to Euler or the complex Barnes integral (expressing the hypergeometric functions as Meijer's $G$-functions). All these can be found in the monographs of W.N. Bailey or L. Slater on hypergeometric functions; alternative sources are Andrews--Askey--Roy and Whittaker--Watson. However, this can never be used in deriving identities/transformations of the hypergeometric functions because of validity issues.</p>