# Hot Integral - 11

**Calculus**Level 5

Let \(\qquad \qquad \displaystyle I_n(x)=\frac{e^{3\pi in/2}2^n}{\sqrt{\pi}}\int\limits_{-\infty}^{\infty}e^{-(t-ix)^2}t^n dt \)

Then, \( \displaystyle \sum_{n=0}^{\infty} \frac{( \frac{2a+1}{2} )_n I_{2n+1}(x) y^{2n}}{\Gamma(2(n+1))} = A(y^B+C)^{- Da - \frac{E}{Q}} x (\;_XF_Y(a+\frac{G}{H} ; \frac{J}{K} ; \frac{y^Lx^M}{y^N+P})) \)

Calculate \(A+B+C+D+E+G+H+J+K+L+M+N+P+Q+X+Y\)

**Details and Assumptions:**

-\(\;_XF_Y(...;.;..)\) represents hypergeometric functions.

-\((b)_n\) represents Pochhammer symbol.

-\(\gcd(E,Q)=\gcd(G,H)=\gcd(J,K)=1\)

-\(A,B,C,D,E,G,H,J,K,L,M,N,P,Q,X,Y\) are all positive integers.

-\(i=\sqrt{-1}\)

\(\blacksquare\) This is a part of Hot Integrals