Toda and Laguerre-Freud equations and tau functions for hypergeometric discrete multiple orthogonal polynomials

In this paper, the authors investigate the case of discrete multiple orthogonal polynomials with two weights on the step line, which satisfy Pearson equations. The discrete multiple orthogonal polynomials in question are expressed in terms of tau functions, which are double Wronskians of generalized hypergeometric series. The shifts in the spectral parameter for type II and type I multiple orthogonal polynomials are described using banded matrices. It is demonstrated that these polynomials offer solutions to multicomponent integrable extensions of the nonlinear Toda equations. Additionally, the paper characterizes extensions of the Nijhoff-Capel totally discrete Toda equations. The hypergeometric $\tau$-functions are shown to provide solutions to these integrable nonlinear equations. Furthermore, the authors explore Laguerre-Freud equations, nonlinear equations for the recursion coefficients, with a particular focus on the multiple Charlier, generalized multiple Charlier, multiple Meixner II, and generalized multiple Meixner II cases.


Introduction
Discrete multiple orthogonal polynomials, -functions, generalized hypergeometric series, Pearson equations, Laguerre-Freud equations, and Toda type equations are intriguing mathematical objects that have attracted considerable attention due to their deep connections with various areas of mathematics and physics.These intertwined concepts play a fundamental role in understanding the underlying structures and dynamics of discrete systems, integrable models, and special functions.In this paper, we delve into the rich mathematical theory of discrete multiple orthogonal polynomials, explore their relationship with -functions and generalized hypergeometric series, and investigate their applications in Laguerre-Freud equations and Toda type equations.
Orthogonal polynomials [45,26,18] are a fascinating and powerful tool in Applied Mathematics and Theoretical Physics, with a wide range of applications in various fields.They possess remarkable properties and have been extensively studied for their applications in approximation theory, numerical analysis, and many other areas.Among the vast family of orthogonal polynomials, two special classes that have attracted significant attention are multiple orthogonal polynomials and discrete orthogonal polynomials.
Back in 1895, Karl Pearson [61], in the context of fitting curves to real data, introduced his famous family of frequency curves by means of the differential equation  ′ (  )  (  ) =  1 (  )  2 (  ) , where  is the probability density and   is a polynomial in  of degree at most ,  = 1, 2. Since then, a vast bibliography has been developed regarding the properties of Pearson distributions.The connection between Pearson equations and orthogonal polynomials lies in the fact that the solutions to Pearson equations often coincide with or are closely related to certain families of orthogonal polynomials.They are characterized by a weight function, which determines the orthogonality properties of the associated orthogonal polynomials.These equations have been extensively studied due to their close relationship with special functions, such as the classical orthogonal polynomials (e.g., Legendre, Hermite, Laguerre) and their generalizations.In this work, we will use the Gauss-Borel approach to orthogonal polynomials, for a review paper see [52].In previous papers, we have explored the application of this technique to the study of discrete hypergeometric orthogonal polynomials, Toda equations, -functions, and Laguerre-Freud equations, see [54,35,36].
Multiple orthogonal polynomials [60,45,56,5] arise when multiple weight functions are involved in the orthogonality conditions.Unlike classical orthogonal polynomials, where a single weight function is used, multiple orthogonal polynomials exhibit a more intricate structure due to the interplay of multiple weight functions.These polynomials have been studied extensively and find applications in areas such as simultaneous approximation, random matrix theory, and statistical mechanics.For the appearance and use of the Pearson equation in multiple orthogonality, see [30,21], see also [22].On the other hand, discrete orthogonal polynomials [45,59] emerge when the orthogonality conditions are defined on a discrete set of points.These polynomials find applications in various discrete systems, such as combinatorial optimization, signal processing, and coding theory.Discrete orthogonal polynomials provide a valuable tool for analyzing and solving problems in discrete settings, where the underlying structure is often characterized by a finite or countable set of points.Discrete Pearson equations play an important role in the classification of classical discrete orthogonal polynomials [59].When a discrete Pearson equation is fulfilled by the weight, we are dealing with semiclassical discrete orthogonal polynomials, see [33,34,29,32].Multiple discrete orthogonal polynomials were discussed in [17].
Integrable discrete equations [44] have emerged as a fascinating and important field of study in Mathematical Physics, providing insights into the behavior of discrete dynamical systems with remarkable properties.These equations possess a rich algebraic and geometric structure, allowing for the existence of soliton solutions, conservation laws, and integrability properties.The concept of integrability in discrete equations goes beyond mere solvability and extends to the existence of an abundance of conserved quantities and symmetries [4].This integrability property enables the development of powerful mathematical methods for studying and understanding their behavior.One of the key features of integrable discrete equations is their connection to the theory of orthogonal polynomials.These polynomials play a central role in the construction of discrete equations with special properties.The link between discrete equations and orthogonal polynomials provides a deep understanding of their solutions, recursion relations, and symmetry properties.
The -function [43] is a key concept in the theory of integrable systems and serves as a bridge between the spectral theory of linear operators and the dynamics of soliton equations.It has profound connections with algebraic geometry, representation theory, and special functions.Semiclassical orthogonal polynomials, along with their corresponding functionals supported on complex curves, have been extensively studied in relation to isomonodromic tau functions and matrix models by Bertola, Eynard, and Harnad [20].In the context of discrete multiple orthogonal polynomials, the -function captures the underlying integrable structure and provides insights into the dynamics and symmetries of the corresponding discrete systems.Its study enables us to uncover deep connections between different mathematical objects and uncover hidden patterns.
Generalized hypergeometric series, another important topic in this research, constitute a class of special functions that arise in diverse mathematical contexts [24,6].They possess remarkable properties and have been extensively studied due to their connections with combinatorics, number theory, and mathematical physics.Generalized hypergeometric series play a key role in expressing solutions of differential equations, evaluating integrals, and understanding the behavior of various mathematical models.They have a pivotal role in the Askey scheme that gathers together different important families of orthogonal families within a chain of limits [46].See [19] for an extension of the Askey scheme to multiple orthogonality.Recently, we have completed a part of the multiple Askey scheme by finding hypergeometric expressions for the Hahn type I multiple orthogonal polynomials and their descendants in the multiple Askey scheme [23] (but for the multiple Hermite case).
Laguerre-Freud equations are also connected to Painlevé equations.The Painlevé equations are a set of nonlinear ordinary differential equations that were first introduced by the French mathematician Paul Painlevé in the early 20th century.These equations have attracted significant attention due to their integrability properties and the rich mathematical structures associated with their solutions.The connection between orthogonal polynomials and Painlevé equations lies in the fact that certain families of orthogonal polynomials can be related to solutions of specific Painlevé equations.This relationship provides deep insights into the solvability and analytic properties of the Painlevé equations and allows for the construction of explicit solutions using the theory of orthogonal polynomials.For an account of the connection between these fields of study, see [64,27,28].
Moreover, building upon the mentioned studies [54,35,36], this paper investigates the relationship between discrete multiple orthogonal polynomials, -functions, generalized hypergeometric series, and specific equations such as Laguerre-Freud equations and Toda type equations.
Toda type equations, on the other hand, are nonlinear partial differential-difference equations with wideranging applications in mathematical physics and integrable systems [43,44].Exploring the connections between these equations and the aforementioned mathematical objects sheds light on the interplay between discrete systems, special functions, and integrable models.
In this paper, we aim to provide a comprehensive overview of the theory of discrete multiple orthogonal polynomials, -functions, generalized hypergeometric series, Laguerre-Freud equations, and Toda type equations.We will delve into the mathematical properties, explicit representations, and recurrence relations of discrete multiple orthogonal polynomials.Furthermore, we will explore the relationship between these polynomials, -functions, and generalized hypergeometric series.Finally, we will investigate the applications of these concepts to Laguerre-Freud equations and Toda type equations, including both continuous and totally discrete types, showcasing their relevance in the field of mathematical physics and integrable systems.
The layout of the paper is as follows.We continue this section with a basic introduction to discrete multiple orthogonal polynomials and their associated tau functions.In the second section, we delve into Pearson equations for multiple orthogonal polynomials, generalized hypergeometric series, and the corresponding hypergeometric discrete multiple orthogonal polynomials.We discuss their properties, including the Laguerre-Freud matrix that models the shift in the spectral variable, as well as contiguity relations and their consequences.
In the third section, we present two of the main findings discussed in this paper.In Theorem 3.9, we present nonlinear partial differential equations of the Toda type, specifically of the third order, for which the hypergeometric tau functions provide solutions.Additionally, in Theorem 3.20, we provide a completely discrete system, extending the completely discrete Nikhoff-Capel Toda equation, and its solutions in terms of the hypergeometric tau functions.
Finally, in the fourth section, we present explicit Laguerre-Freud equations for the first time for the multiple versions of the generalized Charlier and generalized Meixner II orthogonal polynomials.
The moment matrix satisfies Λℳ = ℳ(Λ T ) 2 , so that we say that a matrix is bi-Hankel matrix.The Gauss-Borel or  factorization of the moment matrix is where  and S are lower unitriangular matrices and  is a diagonal matrix.
We define the truncated matrix ℳ [] as the -th leading principal submatrix of the moment matrix ℳ, i.e. built up with the first  rows and columns.Then, we introduce the so called -function, as Factorization (1) exists whenever   ≠ 0 for  ∈ N 0 .
Any diagonal matrix  fulfills the following algebraic properties: Multiple orthogonal polynomials of type II and I on the step-line are given as entries of certain vector of polynomials.Type II multiple orthogonal polynomial   are the entries of the vector , while type I multiple orthogonal polynomials  ()   ,  ∈ {1, 2}, are the entries of the vectors  () where The degrees of these polynomials are easily seen to be deg here ⌈⌉ is the the smallest integer that is greater than or equal to .The following multiple orthogonality relations are satisfied According to [5,Proposition 7] the type II polynomials can be expressed as the following quotient of determinants Hence, for the coefficients    of the type II multiple orthogonal polynomials we get determinantal expressions.For that aim, let us consider the associated -functions    ,  ∈ {1, . . ., −1}, as the determinants of the matrix ℳ [,  ] ,  ∈ {1, . . .,  − 1}, obtained from ℳ [] by removing the ( − )-th row ℳ − ,0 ℳ − ,−1 and adding as last row the row ℳ ,0 ℳ ,−1 .Then, by expanding the determinant in the RHS of (3) by the last column we get Similarly [5] we have The Hessenberg matrix  ≔ Λ −1 and its transposed matrix is  T =  −1 SΛ 2 S−1  give the 4-term recursion relations related to these multiple orthogonal polynomials.Indeed, the following recurrence relations hold Notice that  is a tetradiagonal Hessenberg matrix, i.e., where , , and  are the following diagonal matrices The diagonal matrices , , and  are related with , S and  matrices by the following equations: +  +  [1] =  +  [1] −  [1]  [2] −  [2] −  [1]  − , We introduce the following matrices  (1) and  (2)    () ≔  −1 SΛ () S−1  =  T  () ,  () ≔  −1 S () S−1 ,  ∈ {1, 2}.
as well as  T =  (1) +  (2) .Hence,  () are matrices with all its superdiagonals but for the first and the second have zero entries.The Pascal matrices  = ( , ) have entries given by These matrices act on the monomial vectors as follows and can be expanded as . In terms of Pascal matrices we define the dressed Pascal matrices For type II multiple orthogonal polynomials we find the following properties: We can proceed similarly for the type I multiple orthogonal polynomials.We introduce the matrices  ±() ,  ∈ {1, 2}, with nonzero entries and zero for any other couple sub-indexes.We will also use  () =  +() .These matrices satisfy The partial Pascal matrices can be expressed as band matrices, but in this case the even or odd diagonals are null, i.e., where we have Analogously, dressed Pascal matrices are defined by and the following relations are satisfied  () ( ± 1) = Π ±()  () ().

Proof.
i) This fit in [17,Examples 2.1].ii) This fit in [17,Examples 2.2].□ Proposition 2.5.In terms of generalized hypergeometric functions, the moment matrix can be written as a block matrix where the moments are expressed in terms of generalized hypergeometric series as follows Proof.It follows from the definition of the moment matrix ℳ. Equation ( 15) determining the moments in terms of two families of generalized hypergeometric series follows from Equation ( 14) and [54].□ Given a function  =  () we consider the covector   (  ) ≔     −1  .For two functions  1 ( (1) ) we consider the double Wronskian of the covectors   (  1 ) and   (  2 ) given by the double Wronskian of the covectors  +1 (  1 ) and   (  2 ) given by We refer to these objects as double -Wronskians.Then, the -function (2) can be written as the double -Wronskian of two generalized hypergeometric series as follows (1)    1 , . . .,   ;  (1) , (2)    1 , . . .,   ;  (2) .
Double or 2-component Wronskians were first considered, within the context of integrable systems, by Freeman, Gilson and Nimmo in [41], see also [40].
Lemma 2.6.For the associated -function we find It follows from the multi-linearity of the determinant and Equation ( 14) in where the even columns depend only on  (1) and the odd columns on  (2) .□ Proposition 2.7.The following expressions in terms of the  functions hold true Proof.It can be proven [5] Then, using the previous Lemma 2. 6 we get the result.□ Proposition 2.8.The moment matrix satisfies Proof.It follows from Equations (15).□ Remark 2.9.The previous results apply to more general situations than those provided by the Pearson equation and the generalized hypergeometric series.We only need to assume that  () () =  () ().
Theorem 2.10 (Laguerre-Freud matrix).Let us assume that the weights fulfill discrte Pearson equation (9), where ,  (1) and  (2) are polynomials with deg  =  and max (deg  (1) , deg  (2) ) = .Then, the Laguerre-Freud matrix given by is a banded matrix with lower bandwidth 2 and upper bandwidth , as follows: Moreover, the following connection formulas are fulfilled: so it will have  possible nonzero superdiagonals and using the same argument we conclude that it has 2 subdiagonals possibly nonzero.
We clearly see that ( 45) is an extension to the multiple orthogonal realm of the standard Toda equation.
Next we write this multiple Toda equations as the following Toda system for the recursion coefficients ,  and : Proposition 3.11.For multiple orthogonality with two weights we have the following Toda type system: Proof.Equation ( 46) is immediately obtained from (41a).From (41b), analyzing diagonal by diagonal we can obtain two different equations: [1] = −,  = − [2] + ( −  [1] ) [1] , using equations above and expressions for ,  and  depending of  [2] and  [1] we obtain (46), ( 47) and (48).□ Proposition 3.12 (Lax pair).The matrices  and  are a Lax pair, i.e. the following Lax equation is satisfied , where Proof.We have  = Λ −1 so that and we get Equation (49).The other form can be proved with (41b) and: □ Remark 3.13.As mentioned in Remark 2.9, the previous results are applicable to more general situations beyond generalized hypergeometric series.The only requirement is that  () () =  () ().Once again, we utilize the tau function defined as   =   [ (1)  0 ,  (2)  0 ], where  () 0 = ∞ =0  () ().Coming back to the hypergeometric situation, there is another compatibility equation for Laguerre-Freud matrix, when we make calculations for concrete cases of multiple orthogonal polynomials we will name it Compatibility II.
Proof.We have that Observe that 1 =0 is the vector with all its entries equal to zero but for a 1 in the 2-th entry.Similarly, 1 =0 is the vector with all its entries equal to zero but for a 1 in the (2 + 1)-th entry.Therefore, all the columns of  have 0 all its entries.□ Lemma 3.22.The following equations are satisfied Proof.From eigenvalue equation for  T and  () ,  () T  =   () T , and (34) for this case, Â() () = Hence, if we denote  = ( TT  ( ) − TT  ( ) ) −1 S we find   () = 0 so that, according to Lemma 3.21, we get that  = 0 and as  −1 S is a lower triangular invertible matrix we conclude the result.and finally we get TT  ( ) =  ( )  T .
4. Laguerre-Freud equations for generalized multiple Charlier and Meixner II We will begin by applying the previous developments to the multiple orthogonal polynomials discussed in [17], specifically the multiple Charlier and Meixner II sequences of multiple orthogonal polynomials.It has been proven in [17] that in all these cases, the system is AT, indicating that we are dealing with perfect systems.Subsequently, we will proceed with the study of generalizations of these two cases.The original polynomials were found in [25] and [57], respectively.
The weights are: > 0 and  > −1, according with Lemma 2.4, it is an AT system.
In a recent work [54], we applied this approach to investigate the implications of the Pearson equation on the moment matrix and Jacobi matrices.To describe these implications, we introduced a new banded matrix called the Laguerre-Freud structure matrix, which encodes the Laguerre-Freud relations for the recurrence coefficients.We also discovered that the contiguous relations satisfied generalized hypergeometric functions, determining the moments of the weight described by the squared norms of the orthogonal polynomials.This led to a discrete Toda hierarchy known as the Nijhoff-Capel equation [58].
In [53], we further examined the role of Christoffel and Geronimus transformations in describing the aforementioned contiguous relations and used Geronimus-Christoffel transformations to characterize shifts in the spectral independent variable of the orthogonal polynomials.Building on this program, [35] delved deeper into the study of discrete semiclassical cases, discovering Laguerre-Freud relations for the recursion coefficients of three types of discrete orthogonal polynomials: generalized Charlier, generalized Meixner, and generalized Hahn of type I.
In [36], we completed this program by obtaining Laguerre-Freud structure matrices and equations for three families of hypergeometric discrete orthogonal polynomials.
In this paper, we extend the previous lines of research to multiple orthogonal polynomials on the step line with two weights.We derive hypergeometric -function representations for these multiple orthogonal polynomials and introduce a third-order integrable nonlinear Toda-type equation for these -functions.Additionally, we discover systems of nonlinear discrete Nijhoff-Capel Toda-type equations satisfied by these objects.Finally, we apply the Laguerre-Freud matrix structure and compatibilities to derive Laguerre-Freud equations for the multiple generalized Charlier and multiple generalized Meixner of the second type.
As a future outlook, we highlight the following five lines of research: i) Finding larger families of hypergeometric AT systems.ii) Analyzing the case of  weights with  > 2.
iii) Studying the connections between the topics discussed in this paper and the transformations presented in [21,22] and quadrilateral lattices [31,55].iv) Exploring the relationship with multiple versions of discrete and continuous Painlevé equations involving the coefficients   ,   , and   .The results presented in this paper constitute progress in this direction, as identifying this connection is a nontrivial problem that can be further investigated in future works.v) Discussing higher-order integrable flows and finding applications to multicomponent KP-type equations.