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Sergei
  • Member for 10 years, 8 months
  • Last seen more than a month ago
  • Voronezh, Russia
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closed form expression for an infinite series
It looks like some basic hypergeometric function!
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How to integrate an exponential function of an exponential function?
So the answer is in terms of the function k=1xkk!k. Has it explicit form?
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Why is the Gamma function shifted from the factorial by 1?
But not only two of them. There are much more factorial generalizations which interpolate integer factorials. End even have not poles! For example the Hadamard generalization of factorials, look on Wolfram.
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Inequality for a gamma function
In fact |Γ(s)Γ(2s)|=|Γ(s)Γ(s1)Γ(2s)Γ(s1)|=1π|sinπs||Γ(s)||Γ(s1)|. Due to the well-known inequality |Γ(x+iy)||Γ(x)| both gammas are 1. But why sinus is power-bounded?
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Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions?
I did not find anything about 1F2 in the cited paper of Whipple. It seems there are no such factorization via more simple hypergeometric functions at least for general parameters. Otherwise a problem of asymptotics of 1F2 zeroes would be reduced somehow to something reasonable. But this is unsolved problem as far as I know.
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