Bessel–Clifford function

In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alternative development of the theory of Bessel functions.

satisfies the linear second-order homogeneous differential equation This equation is of generalized hypergeometric type, and in fact the Bessel–Clifford function is up to a scaling factor a Pochhammer–Barnes hypergeometric function; we have Unless n is a negative integer, in which case the right-hand side is undefined, the two definitions are essentially equivalent; the hypergeometric function being normalized so that its value at z = 0 is one.

Similarly, the modified Bessel function of the first kind can be defined as The procedure can of course be reversed, so that we may define the Bessel–Clifford function as but from this starting point we would then need to show

We have, as a special case of Gauss's continued fraction It can be shown that this continued fraction converges in all cases.

The Bessel–Clifford differential equation has two linearly independent solutions.

Since the origin is a regular singular point of the differential equation, and since

, and analytically continue it, we obtain a second linearly independent solution to the differential equation.

We have and In terms of K, we have Hence, just as the Bessel function and modified Bessel function of the first kind can both be expressed in terms of

Collecting terms in t, we find on comparison with the power series definition for