Multiple orthogonal polynomials, $d$-orthogonal polynomials, production matrices, and branched continued fractions

Sokal, Alan D.

doi:10.48550/arxiv.2204.11528

Cited by 3 publications

(6 citation statements)

References 34 publications

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“…The connection between multiple orthogonal polynomials and branched continued fractions was recently introduced and analysed in [37]. Here we revisit and further explore this connection and its applications in the study of multiple orthogonal polynomials.…”

Section: Multiple Orthogonal Polynomials and Branched Continued Fract...mentioning

confidence: 96%

“…This paper gives a detailed investigation of a case study of the connection between two different corners of Mathematics: multiple orthogonal polynomials, studied by the special-functions community, and branched continued fractions, introduced by the enumerative-combinatorics community to solve total-positivity problems. This connection was introduced and analysed in the recent paper [37].…”

Section: Introductionmentioning

confidence: 99%

“…These results were motivated by their applications to the study of multiple orthogonal polynomials, which are made clear in Section 4, but they are worthy of interest on their own. In Section 4, we revisit the connection between multiple orthogonal polynomials and branched continued fractions introduced in [37] and give new contributions to the study of this connection, with emphasis on its application to the analysis of multiple orthogonal polynomials.…”

Section: Introductionmentioning

confidence: 99%

“…Acknowledgements: I am very grateful to Alan Sokal for many enlightening discussions about all the topics of the investigation presented here, especially about the topics concerning lattice paths, branched continued fractions, and total positivity, for numerous pertinent suggestions that considerably improved this paper, and for kindly sharing draft versions of [37] and [11], to Ana Loureiro for illuminating discussions on several results presented here, particularly related to multiple orthogonal polynomials, and to LMS and EPSRC for their support of this work through grants ECF-1920-18 and EP/W522454/1, respectively.…”

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confidence: 99%

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Multiple orthogonal polynomials associated with branched continued fractions for ratios of hypergeometric series

Lima¹

2022

Preprint

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The main objects of the investigation presented in this paper are branched-continued-fraction representations of ratios of contiguous hypergeometric series and type II multiple orthogonal polynomials on the step-line with respect to linear functionals or measures whose moments are ratios of products of Pochhammer symbols. This is an interesting case study of the recently found connection between multiple orthogonal polynomials and branched continued fractions that gives a clear example of how this connection leads to considerable advances on both topics.We start by obtaining new results about generating polynomials of lattice paths and total positivity of matrices and giving new contributions to the general theory of the connection between multiple orthogonal polynomials and branched continued fractions with emphasis on its application to the analysis of multiple orthogonal polynomials. Then, we construct new branched continued fractions for ratios of contiguous hypergeometric series. We give conditions for positivity of the coefficients of these branched continued fractions and we show that the ratios of products of Pochhammer symbols are generating polynomials of lattice paths for a special case of the branched continued fractions under study. Next, we introduce a family of type II multiple orthogonal polynomials on the step-line associated with those branched continued fractions. We present a formula as terminating hypergeometric series for these polynomials, we study their differential properties, and we find an explicit recurrence relation satisfied by them. Finally, we focus the analysis of the multiple orthogonal polynomials to the cases where the corresponding branched-continued-fraction coefficients are all positive. In those cases, the orthogonality conditions can be written using measures on the positive real line involving Meijer G-functions and we obtain results about the location of the zeros and the asymptotic behaviour of the polynomials.Specialisations of the multiple orthogonal polynomials studied here include the classical Laguerre, Jacobi, and Bessel orthogonal polynomials, multiple orthogonal polynomials with respect to Nikishin systems of two measures involving modified Bessel functions, confluent hypergeometric functions, and Gauss' hypergeometric function, and multiple orthogonal polynomials with respect to Meijer Gfunctions used to investigate the singular values of products of Ginibre random matrices as well as a r-orthogonal polynomial sequence with constant recurrence coefficients for any positive integer r and particular instances of the Jacobi-Piñeiro polynomials.

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Section: Multiple Orthogonal Polynomials and Branched Continued Fract...mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Multiple orthogonal polynomials associated with branched continued fractions for ratios of hypergeometric series

Lima¹

2022

Preprint

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“…The multiple orthogonal polynomials associated with weights with no Pochhammer in the denominator were already investigated in [20], where they appeared in the study of products of Ginibre matrices. Weight functions for which the moments are ratios of Pochhammer symbols also appear naturally in the work of Sokal [28], which makes a connection between multiple orthogonal polynomials, production matrices and branched continued fractions. We may extend the multiple orthogonal polynomials in this article to multiple orthogonal polynomials with respect to more weights by considering additional exponential integral weights (w 1 , w 2 , .…”

Section: Related Multiple Orthogonal Polynomialsmentioning

confidence: 99%

Multiple orthogonal polynomials associated with the exponential integral

Assche¹,

Wolfs²

2022

Preprint

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We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights (w 1 , w 2 ) on the positive real line, with w 1 (x) = x α e −x the gamma density and w 2 (x) = x α E ν+1 (x) a density related to the exponential integral E ν+1 . We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider a related family of mixed type multiple orthogonal polynomials.

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Multiple orthogonal polynomials associated with the exponential integral

Assche

Wolfs

2023

Stud Appl Math

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We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights false(w1,w2false)$(w_1,w_2)$ on the positive real line, with w1(x)=xαe−x$w_1(x)=x^\alpha e^{-x}$ the gamma density and w2(x)=xαEν+1(x)$w_2(x) = x^\alpha E_{\nu +1}(x)$ a density related to the exponential integral Eν+1$E_{\nu +1}$. We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series, and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider two related families of mixed‐type multiple orthogonal polynomials.

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Multiple orthogonal polynomials, $d$-orthogonal polynomials, production matrices, and branched continued fractions

Cited by 3 publications

References 34 publications

Multiple orthogonal polynomials associated with branched continued fractions for ratios of hypergeometric series

Multiple orthogonal polynomials associated with branched continued fractions for ratios of hypergeometric series

Multiple orthogonal polynomials associated with the exponential integral

Multiple orthogonal polynomials associated with the exponential integral

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