I would like to learn if there is an orthogonality relation for the hypergeometric functions, namely if I can calculate the integral of the product of two Gaussian Hypergeometric functions $_2F_1$ using the orthogonality relation.
I know that the D.E that the hypergeometric function satisfies is
$$z(1-z) \frac{d^2 w}{dz^2} + (c-(a+b+1) z) \frac{dw}{dz} - a ~ b ~ w = 0 $$
Intuitively I expect that there should be as this is the case for the special functions that are special cases of the hypergeometric function. I don't know if this is the case for each special function, but I am sure for some of them that appear often in physics.
Let me illustrate an example that I can manipulate.
Legendre Polynomials:
This is an example that I understand and can I can calculate the following things.
The orthogonality property of the Legendre polynomials is that the Legendre differential equation can be viewed as a Sturm–Liouville problem, where the Legendre polynomials are eigenfunctions of a Hermitian differential operator, namely
$$\frac{d}{dx}\Bigg((1-x^2)\frac{d P(x)}{dx}\Bigg) = - \lambda P(x)$$
with the eigenvalue $\lambda$ corresponding to $n(n+1)$.
The orthogonality relation, in this case, can be written as
$$\int_{-1}^1 P_m(x) P_n(x) dx = \frac{2}{2n+1} \delta_{m n}$$
The last can be easily proven using Rodrigue’s formula.
Is there a similar formula for the Hypergeometrics?
Thanks in advance.