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Does anyone know anything about the following formula involving special functions: $$ W_{\kappa,\mu}(z)W_{\lambda,\mu}(w)=\frac{e^{-(z+w)/2}(zw)^{\mu+1/2}}{\Gamma(1-\kappa-\lambda)}\int_0^\infty e^{-t}t^{-\kappa-\lambda}(z+t)^{\kappa-\mu-1/2}(w+t)^{\lambda-\mu-1/2} \\ \qquad \qquad \times {}_2F_1\left(\mu-\kappa+1/2,\mu-\lambda+1/2,1-\kappa-\lambda;\frac{t(z+w+t)}{(z+t)(w+t)}\right)\mathrm{d} t \ .\\ \text{for }\qquad \mathrm{Re}(\kappa+\lambda)<1\ ,\qquad z,w\neq 0\ . $$ This formula says that the product of two Whittaker functions $W_{\kappa,\mu}$ is equivalent to an integral of a hypergeometric function ${}_2F_1$ against some weight.

I came across this formula on page 74 of Iwanami mathematical formulas 3 (written in Japanese). You can also find this in equation 7.526.3 of Table of Integrals, Series, and Products, on page 401 of Tables of Integral Transforms volume 2, and in equation 6.15.3.21 in higher transcendental functions. vol. i But I could not find the original paper where this formula is derived. Could you tell me any reference books for this formula?

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I finally found the right paper written by A. Erdélyi in which the formula is introduced. They derived this formula by writing Whittaker functions as Laplace type integral forms and using convolutional properties of Laplace transformation.

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