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Register a new userFurther study on elliptic interpolation formulas for the elliptic Askey-Wilson polynomials and allied identities
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In this paper, we introduce the so-called elliptic Askey-Wilson polynomials which are homogeneous polynomials in two special theta functions. With regard to the significance of polynomials of such kind, we establish some general elliptic interpolation formulas by the methods of matrix
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Motivated by the Gaussian symplectic ensemble, Mehta and Wang evaluated the $n$ by $n$ determinant $det((a+j-i)Gamma(b+j+i))$ in 2000. When $a=0$, Ciucu and Krattenthaler computed the associated Pfaffian $Pf((j-i)Gamma(b+j+i))$ with an application to the two dimensional dimer system in 2011. Recently we have generalized the latter Pfaffian formula with a $q$-analogue by replacing the Gamma function by the moment sequence of the little $q$-Jacobi polynomials. On the other hand, Nishizawa has found a $q$-analogue of the Mehta--Wang formula. Our purpose is to generalize both the Mehta-Wang and Nishizawa formulae by using the moment sequence of the little $q$-Jacobi polynomials. It turns out that the corresponding determinant can be evaluated explicitly in terms of the Askey-Wilson polynomials.
We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials. We also show how to derive a recent double-sum formula for the moments of Askey-Wilson polynomials from Newtons interpolation formula.
An interpolation problem related to the elliptic Painleve equation is formulated and solved. A simple form of the elliptic Painleve equation and the Lax pair are obtained. Explicit determinant formulae of special solutions are also given.
We consider the index problem of certain boundary groupoids of the form $cG = M _0 times M _0 cup mathbb{R}^q times M _1 times M _1$. Since it has been shown that when $q $ is odd and $geq 3$, $K _0 (C^* (cG)) cong bbZ $, and moreover the $K$-theoretic index coincides with the Fredholm index, in this paper we attempt to derive a numerical formula. Our approach is similar to that of renormalized trace of Moroianu and Nistor cite{Nistor;Hom2}. However, we find that when $q geq 3$, the eta term vanishes, and hence the $K$-theoretic and Fredholm indexes of elliptic (respectively fully elliptic) pseudo-differential operators on these groupoids are given only by the Atiyah-Singer term. As for the $q=1$ case we find that the result depends on how the singularity set $M_1$ lies in $M$.
Considering a certain interpolation problem, we derive a series of elliptic difference isomonodromic systems together with their Lax forms. These systems give a multivariate extension of the elliptic Painleve equation.
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