We study differential equations F (y, . . . , y) = 0 where F (Y0, . . . , Yn) is a formal series in Y0, . . . , Yn with coefficients in some field of generalized power series Kr with finite rank r ∈ N ∗. Our purpose is to understand the connection between the set of exponents of the coefficients of the equation Supp F and the set Supp y0 of exponents of the elements y0 ∈ Kr that are solutions.

We prove positivity results about linearization and connection coefficients for Bessel polynomials. The proof is based on a recursion formula and explicit formulas for the coefficients in special cases. The result implies that the distribution of a convex combination of independent Studentt random variables with arbitrary odd degrees of freedom has a density which is a convex combination of cer...

We continue the study of the construction of analytical coefficients of the εexpansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we show the following results: Theorem A: The multiple (inverse) binomial sums

When one wants to use Orthogonal Rational Functions (ORFs) in system identification or control theory, it is important to be able to avoid complex calculations. In this paper we study ORFs whose numerator and denominator polynomial have real coefficients. These ORFs with real coefficients (RORFs) appear when the poles and the interpolation points appear in complex conjugate pairs, which is a na...

If the coefficients, {Aα(x)} do not depend on x, then one can use the Fourier transform, more or less directly to solve this equation. The techniques of microlocal analysis include a calculus which allows one to efficiently quantify and handle the errors which arise from variability of the coefficients. More generally these techniques lead to a precise and quantitative analysis of the singulari...

We give an introduction to the Gauss hypergeometric function, the hypergeometric equation and their properties in an elementary way. Moreover we explicitly and uniformly describe the connection coefficients, the reducibility of the equation and the monodromy group of the solutions.

We continue our study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we apply the approach of obtaining iterated solutions to the differential equations associated with hypergeometric functions to prove the following result: Theorem 1: The epsilon-expansion of a generalized hypergeome...

The authors verify the conjecture that a conjugate pair of zeros can be factored from a polynomial with nonnegative coefficients so that the resulting polynomial still has nonnegative coefficients. The conjecture was originally posed by A. Rigler, S. Trimble, and R. Varga arising out of their work on the Beauzamy-Enflo generalization of Jensen's inequality. The conjecture was also made independ...

In this paper we study some questions in connection with uniform rectifiability and the L boundedness of Calderón-Zygmund operators. We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients which are related to the Jones’ β numbers. We also use these new coefficients to prove that n-dimensional CalderónZygmund operators with odd kernel of type C2 a...

We obtain connection coefficients between q-binomial and q-trinomial coefficients. Using these, one can transform q-binomial identities into a q-trinomial identities and back again. To demonstrate the usefulness of this procedure we rederive some known trinomial identities related to partition theory and prove many of the conjectures of Berkovich, McCoy and Pearce, which have recently arisen in...