Hot Integral - 6

Calculus Level 5

$$\qquad \qquad\displaystyle I_n(x) = \frac{2^n}{\sqrt{\pi}}\int\limits_{-\infty}^{\infty}\frac{(x+it)^n}{e^{t^2}} dt$$

Then, the following summation can be expressed as:

$$\displaystyle \sum_{n=0}^{\infty} \frac{I_n(x)2^n}{\lfloor \frac{n}{2}\rfloor !} = \frac{a+bx}{a^{\frac{c}{d}}}e^{\frac{p}{a}x^q}$$

Calculate $$a+b+c+d+p+q$$.

Details and Assumptions

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

2) $$a,b,c,d,p,q$$ are positive integers.

3) $$\gcd(c,d)=\gcd(p,a)=1$$

4)Sign $$!$$ indicates factorial. For e.g. $$3! = 3 \times 2 \times 1$$

5)$$\lfloor . \rfloor$$ represents the floor function.

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