Applicability of the qq-Analogue of Zeilberger's Algorithm

The applicability or terminating condition for the ordinary case of Zeilberger's algorithm was recently obtained by Abramov. For the $q$-analogue, the question of whether a bivariate $q$-hypergeometric term has a $qZ$-pair remains open. Le has found a solution to this problem when the given bivariate $q$-hypergeometric term is a rational function in certain powers of $q$. We solve the problem for the general case by giving a characterization of bivariate $q$-hypergeometric terms for which the $q$-analogue of Zeilberger's algorithm terminates. Moreover, we give an algorithm to determine whether a bivariate $q$-hypergeometric term has a $qZ$-pair.Comment: 15 page

A Telescoping method for Double Summations

We present a method to prove hypergeometric double summation identities. Given a hypergeometric term $F(n,i,j)$, we aim to find a difference operator $L=a_0(n) N^0 + a_1(n) N^1 +...+a_r(n) N^r$ and rational functions $R_1(n,i,j),R_2(n,i,j)$ such that $L F = \Delta_i (R_1 F) + \Delta_j (R_2 F)$. Based on simple divisibility considerations, we show that the denominators of $R_1$ and $R_2$ must possess certain factors which can be computed from $F(n, i,j)$. Using these factors as estimates, we may find the numerators of $R_1$ and $R_2$ by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews-Paule identity, Carlitz's identities, the Ap\'ery-Schmidt-Strehl identity, the Graham-Knuth-Patashnik identity, and the Petkov\v{s}ek-Wilf-Zeilberger identity.Comment: 22 pages. to appear in J. Computational and Applied Mathematic

The quantum solvation, adiabatic versus nonadiabatic, and Markovian versus non-Markovian nature of electron transfer rate processes

In this work, we revisit the electron transfer rate theory, with particular interests in the distinct quantum solvation effect, and the characterizations of adiabatic/nonadiabatic and Markovian/non-Markovian rate processes. We first present a full account for the quantum solvation effect on the electron transfer in Debye solvents, addressed previously in J. Theore. & Comput. Chem. {\bf 5}, 685 (2006). Distinct reaction mechanisms, including the quantum solvation-induced transitions from barrier-crossing to tunneling, and from barrierless to quantum barrier-crossing rate processes, are shown in the fast modulation or low viscosity regime. This regime is also found in favor of nonadiabatic rate processes. We further propose to use Kubo's motional narrowing line shape function to describe the Markovian character of the reaction. It is found that a non-Markovian rate process is most likely to occur in a symmetric system in the fast modulation regime, where the electron transfer is dominant by tunneling due to the Fermi resonance.Comment: 13 pages, 10 figures, submitted to J. Phys. Chem.