Article history: Received 5 October 2010 Received in revised form 28 March 2011 Accepted 14 April 2011 Available online 20 April 2011

Article history: Received 8 April 2013 Received in revised form 2 June 2014 Accepted 18 June 2014 Available online 8 July 2014 Communicated by H. Prodinger

We derive simple linear, inhomogeneous recurrences for the variance of the index by utilising the fact that the generating function for the distribution of the number of positive eigenvalues of a Gaussian unitary ensemble is a τ-function of the fourth Painlevé equation. From this we deduce a simple summation formula, several integral representations and finally an exact hypergeometric function ...

In this paper we study recurrences concerning the combinatorial sumˆn r

Motivated by recent work of Trümper, we consider the general Collatz word (updown pattern) and the sequences following this pattern. We derive recurrences for the first and last sequence entries from repeated application of the general solution of a binary linear inhomogeneous Diophantine equation. We solve these recurrences and also discuss the Collatz tree.

We give the first comprehensive collection of enumeration results for permutations that avoid barred patterns of length 6 4. We then use the method of prefix enumeration schemes to find recurrences counting permutations that avoid a barred pattern of length > 4 or a set of barred patterns.

In a recent work, M. Covington discusses the enumeration of two different sets of alignments of two strings of symbols using elementary combinatorial techniques. He defines two functions a(m,n) and A(m,n) to count the number of two–string alignments in his “small” and “middle” sets of alignments (respectively). He provides a recurrence for each of these functions which allows for the calculatio...

Quantitative recurrence indicators are defined by measuring the first entrance time of the orbit of a point x in a decreasing sequence of neighborhoods of another point y. It is proved that these recurrence indicators are a.e. greater or equal to the local dimension at y, then these recurrence indicators can be used to have a numerical upper bound on the local dimension of an invariant measure.

In this paper, we define a new kind of the hypergeometric function, say, Hermite-Tricomi functions. An explicit representation, recurrence relations for the Hermite-Tricomi functions are given and differential equations satisfied them is presented. A new expansions of the exponential for a wide class of functions, whose properties, including recurrences, addition theorems, generating functions ...