If a=b=0,m>0,n>0 and \[Rho] < 1
then:
g[m_, n_, \[Rho]_] :=
1/\[Pi] 2^(1/2 (-6 + m + n)) (1 - \[Rho]^2)^(
1/2 (1 + m +
n)) (2 Gamma[(1 + m)/2] Gamma[(1 + n)/2] Hypergeometric2F1[(
1 + m)/2, (1 + n)/2, 1/2, \[Rho]^2] - (
m n Gamma[m/2] Gamma[n/
2] (-(-1 + \[Rho]^2) Hypergeometric2F1[(2 + m)/2, (2 + n)/
2, -(1/2), \[Rho]^2] + (-1 + (4 + m +
n) \[Rho]^2) Hypergeometric2F1[(2 + m)/2, (2 + n)/2, 1/
2, \[Rho]^2]))/((1 + m) (1 + n) \[Rho]))
g[2, 3, 1/2] // N
(* 1.34643*)
f[0, 0, 1/2, 2, 3]
(* 1.34643*)
Regards M.I.
PS. I have another integral:
Integrate[(
E^((-x^2 - y^2 + 2 x y \[Rho])/(2 (1 - \[Rho]^2))) x^m y^n)/(
2 \[Pi] Sqrt[1 - \[Rho]^2]), {x, -Infinity,
Infinity}, {y, -Infinity, Infinity}] ==
1/\[Pi] 2^(1/2 (-4 + m + n)) (1 - \[Rho]^2)^(
1/2 (1 + m + n)) ((
n Gamma[1 + m/2] Gamma[n/
2] (-(-1 + \[Rho]^2) Hypergeometric2F1[(2 + m)/2, (2 + n)/
2, -(1/2), \[Rho]^2] + (-1 + (4 + m +
n) \[Rho]^2) Hypergeometric2F1[(2 + m)/2, (2 + n)/2, 1/
2, \[Rho]^2]) (-1 + Cos[m \[Pi]] + I Sin[m \[Pi]]))/((1 +
m) (1 + n) \[Rho]) +
1/((1 + m) (1 + n) \[Rho]) (-1)^(1 + n)
n Gamma[1 + m/2] Gamma[n/
2] (-(-1 + \[Rho]^2) Hypergeometric2F1[(2 + m)/2, (2 + n)/
2, -(1/2), \[Rho]^2] + (-1 + (4 + m +
n) \[Rho]^2) Hypergeometric2F1[(2 + m)/2, (2 + n)/2, 1/
2, \[Rho]^2]) (-1 + Cos[m \[Pi]] + I Sin[m \[Pi]]) +
Gamma[(1 + m)/2] Gamma[(1 + n)/2] Hypergeometric2F1[(1 + m)/2, (
1 + n)/2, 1/
2, \[Rho]^2] (1 + Cos[m \[Pi]] + I Sin[m \[Pi]]) + (-1)^
n Gamma[(1 + m)/2] Gamma[(1 + n)/2] Hypergeometric2F1[(1 + m)/
2, (1 + n)/2, 1/
2, \[Rho]^2] (1 + Cos[m \[Pi]] + I Sin[m \[Pi]]))
$$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\frac{e^{\frac{-x^2-y^2+2 x y \rho }{2 \left(1-\rho ^2\right)}} x^m
y^n}{2 \pi \sqrt{1-\rho ^2}}dydx=\frac{2^{\frac{1}{2} (-4+m+n)} \left(1-\rho ^2\right)^{\frac{1}{2} (1+m+n)}
\left(\frac{n \Gamma \left(1+\frac{m}{2}\right) \Gamma \left(\frac{n}{2}\right) \left(-\left(-1+\rho ^2\right) \,
_2F_1\left(\frac{2+m}{2},\frac{2+n}{2};-\frac{1}{2};\rho ^2\right)+\left(-1+(4+m+n) \rho ^2\right) \,
_2F_1\left(\frac{2+m}{2},\frac{2+n}{2};\frac{1}{2};\rho ^2\right)\right) (-1+\cos (m \pi )+i \sin (m \pi ))}{(1+m) (1+n)
\rho }+\frac{(-1)^{1+n} n \Gamma \left(1+\frac{m}{2}\right) \Gamma \left(\frac{n}{2}\right) \left(-\left(-1+\rho
^2\right) \, _2F_1\left(\frac{2+m}{2},\frac{2+n}{2};-\frac{1}{2};\rho ^2\right)+\left(-1+(4+m+n) \rho ^2\right) \,
_2F_1\left(\frac{2+m}{2},\frac{2+n}{2};\frac{1}{2};\rho ^2\right)\right) (-1+\cos (m \pi )+i \sin (m \pi ))}{(1+m) (1+n)
\rho }+\Gamma \left(\frac{1+m}{2}\right) \Gamma \left(\frac{1+n}{2}\right) \,
_2F_1\left(\frac{1+m}{2},\frac{1+n}{2};\frac{1}{2};\rho ^2\right) (1+\cos (m \pi )+i \sin (m \pi ))+(-1)^n \Gamma
\left(\frac{1+m}{2}\right) \Gamma \left(\frac{1+n}{2}\right) \, _2F_1\left(\frac{1+m}{2},\frac{1+n}{2};\frac{1}{2};\rho
^2\right) (1+\cos (m \pi )+i \sin (m \pi ))\right)}{\pi }$$