Some Polynomial Sequence Relations

We give some polynomial sequence relations that are generalizations of the Sury-type identities. We provide two proofs, one based on an elementary identity and the other using the method of generating functions.

An Elementary Identity with Cotangent Hyperbolic Function

In this paper, we introduced and proved some important and intresting results on Cotangent Hyperbolic function. Also discussed an important remark at the end of the paper.

In Praise of an Elementary Identity of Euler

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu–Vandermonde sum, the Pfaff–Saalschütz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8\phi_7$ sum. Our proofs are conceptually the same as those obtained by the WZ method, but done without using a computer. We survey identities for Generalized Hypergeometric Series given by Macdonald, and prove several identities for $q$-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new.

An Extension of Foster's Network Theorem

Consider an electrical network onnnodes with resistorsrijbetween nodesiandj. LetRijdenote theeffective resistancebetween the nodes. Then Foster's Theorem [5] asserts thatwherei∼jdenotesiandjare connected by a finiterij. In [10] this theorem is proved by making use of random walks. The classical connection between electrical networks and reversible random walks implies a corresponding statement for reversible Markov chains. In this paper we prove an elementary identity for ergodic Markov chains, and show that this yields Foster's theorem when the chain is time-reversible.We also prove a generalization of aresistive inverseidentity. This identity was known for resistive networks, but we prove a more general identity for ergodic Markov chains. We show that time-reversibility, once again, yields the known identity. Among other results, this identity also yields an alternative characterization of reversibility of Markov chains (see Remarks 1 and 2 below). This characterization, when interpreted in terms of electrical currents, implies thereciprocity theoremin single-source resistive networks, thus allowing us to establish the equivalence ofreversibilityin Markov chains andreciprocityin electrical networks.