Looking for original reference/derivation for expansion of $J_0\left(z\sin\alpha\right)$ in terms of squared Bessel functions and cosines

Question

Several tables list the following equality for the Bessel functions $J_{\nu}\left(x\right)$:

$$J_0\left(z\sin\alpha\right)= J_0\left(\frac{z}{2}\right) + 2\sum_{l=1}^{\infty}J_l^2\left(\frac{z}{2}\right) \cos\left(2l\alpha\right)$$

Examples:

• Magnus, Wilhelm et al. Formulas and Theorems for the Special Functions of Mathematical Physics. DOI: 10.1007/978-3-662-11761-3

referenced by

• Gradshteyn, Izrail Solomonovich et al. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products

I am trying to find the original source of this expression to see how it was derived (I am not looking for a derivation with this question). Magnus, Wilhelm et al. listed above is the oldest source I have found so far.

Answer 1

I traced this expression for $J_0\left(z\sin\alpha\right)$ back to

Carl Neumann. Theorie der Bessel'schen Functionen: Ein Analogon zur Theorie der Kugelfunktionen. B. G. Teubner, Leipzig 1867.

as referenced from

G. N. Watson. A Treatise on the Theory of Bessel Functions, Cambridge University Press.

Actually, the former does not contain the actual expression, but it can be easily constructed from the expressions given and it analyzes the convergence of this series.