Given an n*n matrix M an its rth row and cth column element m[r,c].Suppose m[r,c] = c*m[r-1,c]+a*m[r-1,c-1] and m[0,0]=1, m[0,k]=m[l,0]=0 for k>0,l>0.The letter "a" denotes a constant number and n can be any non negtive integer.

Maple gives the fourier transformation:

fourier(int(y^k*exp(-k*x*y/phi+I*y*omega), y = 0 .. infinity), omega, t) as:

2*Pi*t^k*exp(-t*k*x/phi)*piecewise(t < 0, 0, 1)+piecewise(0 < t, -2*Pi*t^k*exp(-t*k*x/phi), 0)

but it is identically 0.

if change the order of integration and integate exp(I*y*omega) first, the transform is equal to

int(Dirac(y-t)*y^k*exp(-k*x*y/phi), y = 0 .. infinity)

which is Heaviside(t)*t^k*exp(-t*k*x/phi)

s,k,mu,sigma are parameters.k is real number

I want to get a closed form of the nth order differentiation: diff(exp(-(x+sigma)^2/(2*sigma^2)), `$`(x, n)) where sigma is a positive constate. But maple gives an unevaluated answer. Howere, if calculating diff(exp(-x^2/sigma^2), `$`(x, n)),maple gives a closed form:x^(-n)*2^n*MeijerG([[0, 1/2], []], [[0], [1/2+(1/2)*n, (1/2)*n]], x^2/sigma^2).

I have a function

where mu=-19.89674583,sigma=2.35671007 and k=2.475778082 are parameters

I calculate the integration of s with respect of x on intervals of