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Register a new userThe Determinant of an Elliptic Sylvesteresque Matrix
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We evaluate the determinant of a matrix whose entries are elliptic hypergeometric terms and whose form is reminiscent of Sylvester matrices. A hypergeometric determinant evaluation of a matrix of this type has appeared in the context of approximation theory, in the work of Feng, Krattenthaler and Xu. Our determinant evaluation is an elliptic extension of their evaluation, which has two additional parameters (in addition to the base $q$ and nome $p$ found in elliptic hypergeometric terms). We also extend the evaluation to a formula transforming an elliptic determinant into a multiple of another elliptic determinant. This transformation has two further parameters. The proofs of the determinant evaluation and the transformation formula require an elliptic determinant lemma due to Warnaar, and the application of two $C_n$ elliptic formulas that extend Frenkel and Turaevs $_{10}V_9$ summation formula and $_{12}V_{11}$ transformation formula, results due to Warnaar, Rosengren, Rains, and Coskun and Gustafson.
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We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painleve equation (or higher-order analogues), and admitting a large family of monodromy-preserving deformations. The solutions are certain semiclassical biorthogonal functions (and their Cauchy transforms), biorthogonal with respect to higher-order analogues of Spiridonovs elliptic beta integral.
To each nonzero sequence $s:= {s_{n}}_{n geq 0}$ of real numbers we associate the Hankel determinants $D_{n} = det mathcal{H}_{n}$ of the Hankel matrices $mathcal{H}_{n}:= (s_{i + j})_{i, j = 0}^{n}$, $n geq 0$, and the nonempty set $N_{s}:= {n geq 1 , | , D_{n-1} eq 0 }$. We also define the Hankel determinant polynomials $P_0:=1$, and $P_n$, $ngeq 1$ as the determinant of the Hankel matrix $mathcal H_n$ modified by replacing the last row by the monomials $1, x, ldots, x^n$. Clearly $P_n$ is a polynomial of degree at most $n$ and of degree $n$ if and only if $nin N_s $. Kronecker established in 1881 that if $N_s $ is finite then rank $mathcal{H}_{n} = r$ for each $n geq r-1$, where $r := max N_s $. By using an approach suggested by I.S.Iohvidov in 1969 we give a short proof of this result and a transparent proof of the conditions on a real sequence ${t_n}_{ngeq 0}$ to be of the form $t_n=D_n$, $ngeq 0$ for a real sequence ${s_n}_{ngeq 0}$. This is the Hankel determinant problem. We derive from the Kronecker identities that each Hankel determinant polynomial $ P_n $ satisfying deg$P_n = ngeq 1$ is preceded by a nonzero polynomial $P_{n-1}$ whose degree can be strictly less than $n-1$ and which has no common zeros with $ P_n $. As an application of our results we obtain a new proof of a recent theorem by Berg and Szwarc about positive semidefiniteness of all Hankel matrices provided that $D_0 > 0, ldots, D_{r-1} > 0 $ and $D_n=0$ for all $ngeq r$.
We establish discrete and continuous log-concavity results for a biparametric extension of the $q$-numbers and of the $q$-binomial coefficients. By using classical results for the Jacobi theta function we are able to lift some of our log-concavity results to the elliptic setting. One of our main ingredients is a putatively new lemma involving a multiplicative analogue of Turans inequality.
We give elementary proofs of the univariate elliptic beta integral with bases $|q|, |p|<1$ and its multiparameter generalizations to integrals on the $A_n$ and $C_n$ root systems. We prove also some new unit circle multiple elliptic beta integrals, which are well defined for $|q|=1$, and their $pto 0$ degenerations.
We give a brief account of the key properties of elliptic hypergeometric integrals -- a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic generalization of Eulers and Selbergs beta integrals, elliptic analogue of the Euler-Gauss hypergeometric function and some multivariable elliptic hypergeometric functions on root systems. The elliptic Fourier transformation and corresponding integral Bailey lemma technique is outlined together with a connection to the star-triangle relation and Coxeter relations for a permutation group. We review also the interpretation of elliptic hypergeometric integrals as superconformal indices of four dimensional supersymmetric quantum field theories and corresponding applications to Seiberg type dualities.
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