This paper discusses analysis of recursive problems. It delves first on their classification, and then the various methods of solving them, depending on which class the recursive relation belongs to. An improvement on the Master Method is then described and used to demonstrate how this method is used to solve recursive relations on Divide & Conquer problems. The revised method is found easier t...

Somos 4 sequences are a family of sequences defined by a fourthorder quadratic recurrence relation with constant coefficients. For particular choices of the coefficients and the four initial data, such recurrences can yield sequences of integers. Fomin and Zelevinsky have used the theory of cluster algebras to prove that these recurrences also provide one of the simplest examples of the Laurent...

In 1858, Stern introduced an array, later called the diatomic array. The array is formed by taking two values a and b for the first row, and each succeeding row is formed from the previous by inserting c+d between two consecutive terms with values c, d. This array has many interesting properties, such as the largest value in a row of the diatomic array is the (r + 2)-th Fibonacci number, occurr...

This paper will prove that essentially only the obvious recurrences have almost all primes as divisors. An integer n is a divisor of a recurrence if n divides some term of the recurrence. In this paper, "almost all primes" will be taken interchangeably to mean either all but finitely many primes or all but for a set of Dirichlet density zero in the set of primes. In the context of this paper, t...

Using integration by parts relations, Feynman integrals can be represented in terms of coupled systems of differential equations. In the following we suppose that the unknown Feynman integrals can be given in power series representations, and that sufficiently many initial values of the integrals are given. Then there exist algorithms that decide constructively if the coefficients of their powe...

ABSTRACT We consider systems Al(t)y(q t)+ . . . +A0(t)y(t) = b(t) of higher order q-recurrence equations with rational coecients. We extend a method for €nding a bound on the maximal power of t in the denominator of arbitrary rational solutionsy(t) aswell as amethod for bounding the degree of polynomial solutions from the scalar case to the systems case. Œe approach is direct and does not rely...

We study the existence of periodic solutions of the non–autonomous periodic Lyness’ recurrence un+2 = (an +un+1)/un, where {an}n is a cycle with positive values a,b and with positive initial conditions. It is known that for a = b = 1 all the sequences generated by this recurrence are 5–periodic. We prove that for each pair (a, b) 6= (1, 1) there are infinitely many initial conditions giving ris...

We prove new double summation hypergeometric q-series representations for several families of partitions, including those that appear in the famous product identities of Göllnitz, Gordon, and Schur. We give several different proofs for our results, using bijective partitions mappings and modular diagrams, the theory of q-difference equations and recurrences, and the theories of summation and tr...

In this article, we present a general solution for linear divide-and-conquer recurrences of the form