Parametrized Stochastic Grammars for RNA Secondary Structure Prediction

We propose a two-level stochastic context-free grammar (SCFG) architecture for parametrized stochastic modeling of a family of RNA sequences, including their secondary structure. A stochastic model of this type can be used for maximum a posteriori estimation of the secondary structure of any new sequence in the family. The proposed SCFG architecture models RNA subsequences comprising paired bases as stochastically weighted Dyck-language words, i.e., as weighted balanced-parenthesis expressions. The length of each run of unpaired bases, forming a loop or a bulge, is taken to have a phase-type distribution: that of the hitting time in a finite-state Markov chain. Without loss of generality, each such Markov chain can be taken to have a bounded complexity. The scheme yields an overall family SCFG with a manageable number of parameters.Comment: 5 pages, submitted to the 2007 Information Theory and Applications Workshop (ITA 2007

Extensions of the Classical Transformations of 3F2

It is shown that the classical quadratic and cubic transformation identities satisfied by the hypergeometric function ${}_3F_2$ can be extended to include additional parameter pairs, which differ by integers. In the extended identities, which involve hypergeometric functions of arbitrarily high order, the added parameters are nonlinearly constrained: in the quadratic case, they are the negated roots of certain orthogonal polynomials of a discrete argument (dual Hahn and Racah ones). Specializations and applications of the extended identities are given, including an extension of Whipple's identity relating very well poised ${}_7F_6(1)$ series and balanced ${}_4F_3(1)$ series, and extensions of other summation identities.Comment: 22 pages, expanded version, to appear in Advances in Applied Mathematic