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Could anyone suggest a reference (book or paper, no URL though) for this identity involving the ${}_2F_{2}$ hypergeometric function?

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Hypergeometric functions $_pF_q$ whose parameters differ by integers are said to be contiguous; if $p\leq q+1$ then any $q+2$ distinct contiguous functions are linearly related. See the references listed in DLMF entry 16.3(ii).

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  • $\begingroup$ I have found that many of the contiguous relations can be directly computed by "resolving" (eliminating the derivate of qFp between instances) the expanded (n=1) relations dlmf.nist.gov/16.3.i . Which in turn can be computed by examing the individual power series expansions of the terms. This is not the totality but shows some cases of contiguous relations are general; not specialized. There are specialized ones that would be hard to get to this way though. $\endgroup$ – rrogers Mar 7 '18 at 14:35

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