In this paper, we consider a two-dimensional model for finite set partitions which arises in conjunction with a special case of a general non-linear recurrence. We investigate properties of some of the related counting sequences, including recurrences and generating functions. In particular, we obtain, by combinatorial arguments, some formulas relating these sequences to the Stirling numbers of...

In recent analyses [3, 4] the remarkable AGM continued fraction of Ramanujan—denoted R1(a, b)—was proven to converge for almost all complex parameter pairs (a, b). It was conjectured that R1 diverges if and only if (0 = a = be with cos2 φ = 1) or (a2 = b2 ∈ (−∞, 0)). In the present treatment we resolve this conjecture to the positive, thus establishing the precise convergence domain for R1. Thi...

The polynomial invariants qd for a large class of smooth 4-manifolds are shown to satisfy universal relations. The relations reflect the possible genera of embedded surfaces in the 4-manifold and lead to a structure theorem for the polynomials. As an application, one can read off a lower bound for the genera of embedded surfaces from the asymptotics of qd for large d. The relations are proved u...

A sequence of graphs Gn is iteratively constructible if it can be built from an initial labeled graph by means of a repeated fixed succession of elementary operations involving addition of vertices and edges, deletion of edges, and relabelings. Let Gn be a iteratively constructible sequence of graphs. In a recent paper, [27], M. Noy and A. Ribò have proven linear recurrences with polynomial coe...

This note compares and contrasts some properties of binary sequences with matrices and associated recurrence relations in order to stimulate some enrichment exercises and pattern puzzles.

The most efficient algorithms to evaluate Feynman diagrams exploit recursive methods based on integration by parts technique [2] or technique of generalized recurrence relations [3], [4]. It is easy to derive hundreds of recurrence relations but it is by far not so easy to formulate optimal algorithm how to use them for the reduction of a given type of integrals to a minimal set of basis integr...

We calculate certain features of Bose–Einstein condensation in the ideal gas by using recurrence relations for the partition function. The grand canonical ensemble gives inaccurate results for certain properties of the condensate that are accurately provided by the canonical ensemble. Calculations in the latter can be made tractable for finite systems by means of the recurrence relations. The i...

In this paper, we establish some new recurrence relations between the single moments of record values from the modified Weibull distribution as well as between the double moments. Next, we show that the results for the record values from the Weibull distribution obtained by Balakrishnan and Chan (1993) can be derived from our results as special cases. In addition, we deduce some new recurrence ...

We show that the problem of solving recurrence relations for L-loop (R + 1)-point Feynman integrals within the method of integration by parts is equivalent to the corresponding problem for (L + R)-loop vacuum or (L + R − 1)-loop propagator-type integrals. Using this property we solve recurrence relations for two-loop massless vertex diagrams, with arbitrary numerators and integer powers of prop...