# Question:Could the parametric Identity for Sqrt(Pi) be be simplified and/or proven?

## Question:Could the parametric Identity for Sqrt(Pi) be be simplified and/or proven?

Maple

Hi,

Below is the parametric identity, which i have found empirically: sqrt[pi] =(1/(2^j) ((k*gamma[5 + 2 j] gamma[ 1 + l] hypergeometricpfq[{1, 5/2 + j, 3 + j}, {3 + j + l/2, 7/2 + j + l/2}, -1])/ gamma[6 + 2 j + l] + ((k + m) gamma[7 + 2 j] gamma[ 1 + l] hypergeometricpfq[{1, 7/2 + j, 4 + j}, {4 + j + l/2, 9/2 + j + l/2}, -1])/gamma[8 + 2 j + l]))/(2^(-5 - 3 j - l) gamma[ 5 + 2 j] gamma[ 1 + l] (k hypergeometricpfqregularized[{1, 5/2 + j, 3 + j}, {3 + j + l/2, 7/2 + j + l/2}, -1] + 1/2 (3 + j) (5 + 2 j) (k + m) hypergeometricpfqregularized[{1, 7/2 + j, 4 + j}, {4 + j + l/2, 9/2 + j + l/2}, -1]

It seems to be true for arbitrary j,k,l,m parameters where j, k, l and m are signed integer

For all tried specific sets of {j,k,l,m} above identity was confirmed by both mathematica based wolframalpha and maple.

Could this identity be simplified?

How this identity could be proven either by using mathematica and/or analyticall

Thanks,

Best regards,

Alexander R. Povolotsky

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