This paper surveys examples in which questions are answered by developing recurrence relations that model certain situations and then exploring those recurrence relations using an electronic spreadsheet. The goals of this approach are (1) to develop an understanding of and a facility for using recurrence relations as a modeling tool to motivate the need to derive analytical methods for solving ...

We describe the origins of recurrence relations between field theory amplitudes in terms of the construction of Feynman diagrams. In application we derive recurrence relations for the amplitudes of QED which hold to all loop orders and for all combinations of external particles. These results may also be derived from the Schwinger-Dyson equations.

It is well known that members of families of polynomials, that are orthogonal with respect to an inner product determined by a nonnegative measure on the real axis, satisfy a three-term recursion relation. Analogous recursion formulas are available for orthogonal Laurent polynomials with a pole at the origin. This paper investigates recursion relations for orthogonal rational functions with arb...

The convergence of new second-order iterative methods are analyzed in Banach spaces by introducing a system of recurrence relations. A system of a priori error bounds for that method is also provided. The methods are defined by using a constant bilinear operator A, instead of the second Fréchet derivative appearing in the defining formula of the Chebyshev method. Numerical evidence that the met...

Explicit formulas for solutions of recurrence relations for 3–loop vacuum integrals are generalized for the n-loop case. Recently [1] a new approach to implement recurrence relations [2] for vacuum integrals was suggested. Such relations connect Feynman integrals with various degrees of their denominators. In many cases they provide a possibility to express an integral with given degrees of den...

In a previous paper, we showed that in the semiclassical (WKB) Coulomb approximation, the radial integrals for mu)tipole transitions between nonhydrogenic atomic states obey simple recurrence relations. This paper deals with the recurrence relations connecting the nonhydrogenic radial matrix elements expressed in the quantal form. Numerical tests have been made and give good results.

A non-decreasing sequence of positive integers is (α, β)-Conolly, or Conollylike for short, if for every positive integer m the number of times that m occurs in the sequence is α + βrm, where rm is 1 plus the 2-adic valuation of m. A recurrence relation is (α, β)-Conolly if it has an (α, β)-Conolly solution sequence. We discover that Conolly-like sequences often appear as solutions to nested (o...

In a recently published book, it is shown that there are essentially seven different non– trivial systems of three intertwined second–order recurrence relations, but the solutions are not given. In this note we present explicit solutions to all seven systems. AMS Classification: 11B39, 11B37

We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers. Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial “counting” all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials)...