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Register a new userGeneralizing Krawtchouk polynomials using Hadamard matrices
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We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged as a matrix, where q is a positive integer greater than one. Orthogonality relations are established and coefficients are obtained for the expansion of a polynomial in terms of m-polynomials. We conclude this article by an implementation in MATHEMATICA of m-polynomials and the results obtained for them.
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In this paper, we generalize classical constructions of skew Hadamard difference families with two or four blocks in the additive groups of finite fields given by Szekeres (1969, 1971), Whiteman (1971) and Wallis-Whiteman (1972). In particular, we show that there exists a skew Hadamard difference family with $2^{u-1}$ blocks in the additive group of the finite field of order $q^e$ for any prime power $qequiv 2^u+1,({mathrm{mod, , }2^{u+1}})$ with $uge 2$ and any positive integer $e$. In the aforementioned work of Szekeres, Whiteman, and Wallis-Whiteman, the constructions of skew Hadamard difference families with $2^{u-1}$ ($u=2$ or $3$) blocks in $({mathbb F}_{q^e},+)$ depend on the exponent $e$, with $eequiv 1,2,$ or $3,({mathrm{mod, , }4})$ when $u=2$, and $eequiv 1,({mathrm{mod, , }2})$ when $u=3$, respectively. Our more general construction, in particular, removes the dependence on $e$. As a consequence, we obtain new infinite families of skew Hadamard matrices.
If $q = p^n$ is a prime power, then a $d$-dimensional emph{$q$-Butson Hadamard matrix} $H$ is a $dtimes d$ matrix with all entries $q$th roots of unity such that $HH^* = dI_d$. We use algebraic number theory to prove a strong constraint on the dimension of a circulant $q$-Butson Hadamard matrix when $d = p^m$ and then explicitly construct a family of examples in all possible dimensions. These results relate to the long-standing circulant Hadamard matrix conjecture in combinatorics.
In this paper, we obtain a number of new infinite families of Hadamard matrices. Our constructions are based on four new constructions of difference families with four or eight blocks. By applying the Wallis-Whiteman array or the Kharaghani array to the difference families constructed, we obtain new Hadamard matrices of order $4(uv+1)$ for $u=2$ and $vin Phi_1cup Phi_2 cup Phi_3 cup Phi_4$; and for $uin {3,5}$ and $vin Phi_1cup Phi_2 cup Phi_3$. Here, $Phi_1={q^2:qequiv 1pmod{4}mbox{ is a prime power}}$, $Phi_2={n^4in mathbb{N}:nequiv 1pmod{2}} cup {9n^4in mathbb{N}:nequiv 1pmod{2}}$, $Phi_3={5}$ and $Phi_4={13,37}$. Moreover, our construction also yields new Hadamard matrices of order $8(uv+1)$ for any $uin Phi_1cup Phi_2$ and $vin Phi_1cup Phi_2 cup Phi_3$.
S-Heun operators on linear and $q$-linear grids are introduced. These operators are special cases of Heun operators and are related to Sklyanin-like algebras. The Continuous Hahn and Big $q$-Jacobi polynomials are functions on which these S-Heun operators have natural actions. We show that the S-Heun operators encompass both the bispectral operators and Kalnins and Millers structure operators. These four structure operators realize special limit cases of the trigonometric degeneration of the original Sklyanin algebra. Finite-dimensional representations of these algebras are obtained from a truncation condition. The corresponding representation bases are finite families of polynomials: the para-Krawtchouk and $q$-para-Krawtchouk ones. A natural algebraic interpretation of these polynomials that had been missing is thus obtained. We also recover the Heun operators attached to the corresponding bispectral problems as quadratic combinations of the S-Heun operators
Entrywise powers of matrices have been well-studied in the literature, and have recently received renewed attention in the regularization of high-dimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices $(H_{st})_{s,t=1}^n$ with complex entries. We first characterize the powers $alphainmathbb{R}$ such that the blockwise power map $(H_{st}) mapsto (H_{st}^alpha)$ preserves Loewner positivity. The characterization is obtained by exploiting connections with the theory of matrix monotone functions developed by Loewner. Second, we revisit previous work by Choudhury [Proc. AMS 108] who had provided a lower bound on $alpha$ for preserving positivity when the blocks $H_{st}$ pairwise commute. We completely settle this problem by characterizing the full set of powers preserving positivity in this setting. Our characterizations generalize previous work by FitzGerald-Horn, Bhatia-Elsner, and Hiai from scalars to arbitrary block size, and in particular, generalize the Schur Product Theorem. Finally, a natural and unifying framework for studying the case of diagonalizable blocks consists of replacing real powers by general characters of the complex plane. We thus classify such characters, and generalize our results to this more general setting. In the course of our work, given $betainmathbb{Z}$, we provide lower and upper bounds for the threshold power $alpha >0$ above which the complex characters $re^{itheta}mapsto r^alpha e^{ibetatheta}$ preserve positivity when applied entrywise to positive semidefinite matrices. In particular, we completely resolve the $n=3$ case of a question raised in 2001 by Xingzhi Zhan. As an application, we extend previous work by de Pillis [Duke Math. J. 36] by classifying the characters $K$ of the complex plane for which the map $(H_{st})_{s,t=1}^n mapsto (K({rm tr}(H_{st})))_{s,t=1}^n$ preserves positivity.
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