A holonomic systems approach to special functions identities

Abstract

We observe that many special functions are solutions of so-called holonomic systems. Bernstein's deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps. This is partially substantiated by an algorithm that proves terminating hypergeometric series identities, and that is given both in English and in MAPLE.

Computer Algebra Meets Finite Elements: An Efficient Implementation for Maxwell’s Equations

“…The methods that we employ, originate in Zeilberger's holonomic systems approach [13,3,10] whose basic idea is to define functions and sequences in terms of differential equations and recurrence equations plus initial values (these equations have to be linear with polynomial coefficients). Luckily the shape functions used in the chosen FEM discretization fit into the holonomic framework since they are defined in terms of orthogonal polynomials.…”

Structural theorems for symbolic summation

“…This framework and extensions [42,43,44,23,48,24,45,25] generalize, e.g., the (q-)hypergeometric algorithms presented in [1,18,54,34,32,35,33,5,20,3], they cover as special case the summation of (q-)harmonic sums [10,51,29,11] arising, e.g., in particle physics, and they can treat classes of multi-sums that are out of scope of, e.g., the holonomic approach [53,52,15,14]. Karr's algorithm can be considered as the discrete analogue of Risch's algorithm [36,37] for indefinite integration.…”

On the average complexity for the verification of compatible sequences

“…It satisfies a linear recurrence in n with polynomial coefficients and as such it fits into the so-called holonomic framework [22,1,16]. Within the last decades several symbolic algorithms have been designed and implemented that are capable of automatically finding (and thus proving!)…”