We define a new difference equation analogue of the Bessel differential equation and investigate the properties of its solution, which we express using a
2
F
1
{}_2F_1
hypergeometric function. We find analogous formulas for Bessel function recurrence relations, a summation transformation which is identical to the Laplace transform of classical Bessel functions, and oscillation.
A difference equation analogue of the generalized hypergeometric differential equation is defined, its contiguous relations are developed, and its relation to numerous well-known classical special functions are demonstrated.
We provide proofs of sharp divergence criteria for a large class of matrix hypergeometric series by estimating lower norm bounds of their terms. Difficulties in extending our techniques to singular matrix parameters are illustrated.
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