Can the following sum be counted or expressed in terms of special functions?

integration – fractional moments of multivariate regular distributions Answer

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integration – fractional moments of multivariate regular distributions

Mathematica offers some outcomes for the bivariate regular distribution with correlation $rho$ and benchmark marginals. If $a$ and $b$ are constructive integers then

commence{align}
textual content{for queer }a+b, E[x^a y^b] &=0

textual content{for queer }atext{ and }b, E[x^a y^b] &=
sqrt{frac{2^{a+1}}pi}
Gamma!left(1+frac a2right)
f(a,b,rho)

textual content{for plane }atext{ and }b, E[x^a y^b] &=
sqrt{frac{2^{a}}pi}
Gamma!left(1+frac a2right)
f(a,b,rho)
aim{align}
the place
commence{align}
f(a,1,rho)&=rho
f(a,3,rho)&=3rho+(a-1)rho^3
f(a,5,rho)&=15rho+10(a-1)rho^3+(a-1)(a-3)rho^5
f(a,2,rho)&=1+arho^2
f(a,4,rho)&=3+6arho^2+a(a-2)rho^4
f(a,6,rho)&=15+45arho^2+15a(a-2)rho^4 + a(a-2)(a-4)rho^6
aim{align}

For queer $a$ and $b$ the coefficients emerge to breathe Ward numbers, and for plane $a$ and $b$ they emerge to breathe the exponential Riordan array.

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