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. If

is the entire function defined by means of the reciprocal Gamma function, then the Bessel–Clifford function is defined by the series

The ratio of successive terms is z/k(n + k), which for all values of z and n tends to zero with increasing k. By the ratio test, this series converges absolutely for all z and n, and uniformly for all regions with bounded |z|, and hence the Bessel–Clifford function is an entire function of the two complex variables n and z.

It follows from the above series on differentiating with respect to x that ${\displaystyle {\mathcal {C}}_{n}(x)}$ 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

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