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 (1930–1987).
[1][2][3] For a positive integer
, the Narayana number
For a nonnegative integer
is defined by The associated Narayana polynomial
is defined as the reciprocal polynomial of
: The first few Narayana polynomials are 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.
The Narayana polynomials can be expressed in the following alternative form:[4] The ordinary generating function the Narayana polynomials is given by The
-th degree Legendre polynomial
can be expressed in the following form: