The Art of Bijective Combinatorics Part IV

Combinatorial theory of orthogonal polynomials and continued fractions

The Institute of Mathematical Sciences, Chennai, India (January-March 2019)

Some possible complementary chapters

Some possible complementary chapters:

Chapter 7 Linearization coefficients

Linearization coefficients of the five Scheffer orthogonal polynlomials, combinatorial interpretation

Linearization coefficiens of q-Hermite, q-Charlier and q-Laguerre polynomials

Chapter 8 Operators, quadratic algebra and orthogonal polynomials

q-Hermite polynomials and the (Weyl-Heisenberg) quadratic algebra defined by UD = qDU + Id

Rook placements, q-Hermite, operators U,D and Al-Salam--Chihara polynomials

q-Laguerre polynomials and the (PASEP) quadratic algebra defined by DE = qED + E + D

Chapter 9 Applications and interactions

The birth and death process in probability theory revisited with the combinatorics of orthogonal polynomials

Computing integrated cost for data structures in computer science:

stack, priority queue, dictionnary, linear list, symbol table

The PASEP model in physics and its matrix ansatz

Orthogonal polynomials and Smith normal form

Chapter 10 Extensions

Biorthogonality

L-fractions, extension of the matrix inversion theorem between orthogonal polynomials and inverse (= vertical) polynomials.

multicontinued fractions, T-fractions, tree-like continued fractions, some examples

Combinatorial theory of Padé approximants and P-fractions