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Is there any identity in the confluent hypergeometric function that connects the following? ${}_1{F_1}\left( {a + n + 1,b + n + 1, - c} \right)$ and ${}_1{F_1}\left( {a + 1,b + 1, - c} \right)$ where a, b and c are positive real numbers

user827039
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1 Answers1

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Note that $$ \frac{d}{dx}\;{}_1F_1(a,b,x) = \frac{a}{b}\;{}_1F_1(a+1,b+1,x) $$ Apply this inductively to get $$ \frac{d^n}{dc^n}\;{}_1F_1(a+1,b+1,-c) = (-1)^n\;\frac{(a+1)(a+2)\cdots(a+n)}{(b+1)(b+2)\cdots(b+n)} {}_1F_1(a+n+1,b+n+1,-c) $$

GEdgar
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