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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.

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