Narayana polynomials
Narayana polynomials are a class of polynomials whose coefficients are the Narayana numbers. The Narayana numbers and Narayana polynomials are named after the Canadian mathematician T. V. Narayana. They appear in several combinatorial problems.
Definitions
For a positive integer and for an integer, the Narayana number is defined byThe number is defined as for and as for.
For a nonnegative integer, the -th Narayana polynomial is defined by
The associated Narayana polynomial is defined as the reciprocal polynomial of :
Examples
The first few Narayana polynomials areProperties
A few of the properties of the Narayana polynomials and the associated Narayana polynomials are collected below. Further information on the properties of these polynomials are available in the references cited.Alternative form of the Narayana polynomials
The Narayana polynomials can be expressed in the following alternative form:Special values
- is the -th Catalan number. The first few Catalan numbers are..
- is the -th large Schröder number. This is the number of plane trees having edges with leaves colored by one of two colors. The first few Schröder numbers are..
- For integers, let denote the number of underdiagonal paths from to in a grid having step set. Then.
Recurrence relations
- For, satisfies the following nonlinear recurrence relation:
- For, satisfies the following second order linear recurrence relation:
Generating function
The ordinary generating function the Narayana polynomials is given byIntegral representation
The -th degree Legendre polynomial is given byThen, for n > 0, the Narayana polynomial can be expressed in the following form:
- .