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Suppose n(x) was defined somehow (in fact, in extenso) but see also my related question about how to import Maxima built-in/package functions. My current result so far:

absm(p)=integral(abs(x)^p*n(x), x, -infinity, infinity)
absm
    p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
a(k,p)=integral(integral(abs(u)^p*abs(u+sqrt(k-1)*v)^p*n(v)*n(u), v, -infinity, infinity), u, -infinity, infinity)
a(k,p)
    1/2*integrate(abs(u)^p*integrate(abs(sqrt(k - 1)*v + u)^p*e^(-1/2*u^2 - 1/2*v^2), v, -Infinity, +Infinity), u, -Infinity, +Infinity)/pi
ab(p)=a(1,p/2).simplify()
ab
    p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
bool(absm == ab)
    True

This is a partial result but a(k,p) can't be simplified in terms of gamma's or whatnots.

Suppose n(x) was defined somehow (in fact, in extenso) but see also my related question about how to import Maxima built-in/package functions. My current result so far:

absm(p)=integral(abs(x)^p*n(x), x, -infinity, infinity)
absm
    p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
a(k,p)=integral(integral(abs(u)^p*abs(u+sqrt(k-1)*v)^p*n(v)*n(u), v, -infinity, infinity), u, -infinity, infinity)
a(k,p)
    1/2*integrate(abs(u)^p*integrate(abs(sqrt(k - 1)*v + u)^p*e^(-1/2*u^2 - 1/2*v^2), v, -Infinity, +Infinity), u, -Infinity, +Infinity)/pi
ab(p)=a(1,p/2).simplify()
ab
    p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
bool(absm == ab)
    True

This is a partial result but a(k,p) can't be simplified in terms of gamma's or whatnots.

Suppose n(x) was defined somehow (in fact, in extenso) but see also my related question about how to import Maxima built-in/package functions. My current result so far:

absm(p)=integral(abs(x)^p*n(x), x, -infinity, infinity)
absm
    p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
a(k,p)=integral(integral(abs(u)^p*abs(u+sqrt(k-1)*v)^p*n(v)*n(u), v, -infinity, infinity), u, -infinity, infinity)
a(k,p)
    1/2*integrate(abs(u)^p*integrate(abs(sqrt(k - 1)*v + u)^p*e^(-1/2*u^2 - 1/2*v^2), v, -Infinity, +Infinity), u, -Infinity, +Infinity)/pi
ab(p)=a(1,p/2).simplify()
ab
    p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
bool(absm == ab)
    True

This is a partial result but a(k,p) can't be simplified in terms of gamma's or whatnots.

How do you make changes of variables here?in Sage?