Multiple orthogonal polynomials, \(d\)-orthogonal polynomials, production matrices, and branched continued fractions. (English) Zbl 1536.42027
Summary: I analyze an unexpected connection between multiple orthogonal polynomials, \(d\)-orthogonal polynomials, production matrices and branched continued fractions. This work can be viewed as a partial extension of Viennot’s combinatorial theory of orthogonal polynomials to the case where the production matrix is lower-Hessenberg but is not necessarily tridiagonal.
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 05A15 | Exact enumeration problems, generating functions |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 15A24 | Matrix equations and identities |
| 15A99 | Basic linear algebra |
| 15B99 | Special matrices |
| 30B70 | Continued fractions; complex-analytic aspects |
| 30E05 | Moment problems and interpolation problems in the complex plane |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 41A21 | Padé approximation |
Keywords:
multiple orthogonal polynomial; \(d\)-orthogonal polynomial; production matrix; branched continued fraction; lower-Hessenberg matrix; Łukasiewicz pathSoftware:
OEISReferences:
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