Korean J. Math. Vol. 25 No. 2 (2017) pp.181-199
DOI: https://doi.org/10.11568/kjm.2017.25.2.181

The recurrence coefficients of the orthogonal polynomials with the weights $w_\alpha(x)= x^\alpha \exp(-x^3+tx)$ and $W_\alpha(x)=|x|^{2\alpha+1} \exp(-x^6+tx^2)$

Main Article Content

Haewon Joung


In this paper we consider the orthogonal polynomials with weights $w_\alpha(x)= x^\alpha \exp(-x^3+tx)$ and $W_\alpha(x)=|x|^{2\alpha+1} \exp(-x^6+tx^2)$. Using the compatibility conditions for the ladder operators for these orthogonal polynomials, we derive several difference equations satisfied by the recurrence coefficients of these orthogonal polynomials. We also derive differential-difference equations and second order linear ordinary differential equations satisfied by these orthogonal polynomials.

Article Details


[1] Boelen L, Filipuk G, and Van Assche W, Recurrence coefficients of generalized Meixner polynomials and Painlev e equations, J. Phys. A: Math. Theor. 44 (2011) 035202. Google Scholar

[2] Bonan S and Clark D S, Estimates of the orthogonal polynomials with weight $exp(-x^m)$, $m$ an even positive integer, J. Approx. Theory 46 (1986), 408–410. Google Scholar

[3] Chen Y and Ismail M, Jacobi polynomials from compatibility conditions, Proc. Am. Math. Soc. 133 (2005), 465–472. Google Scholar

[4] Chihara T S, An introduction to orthogonal polynomials, Gordon and Breach, New york 1978. Google Scholar

[5] Filipuk G, Van Assche W, and Zhang L, The recurrence coefficients of semi- classical Laguerre polynomials and the fourth Painlev e equation, J. Phys. A: Math. Theor. 45 (2012) 205201. Google Scholar

[6] Shohat J, A differential equation for orthogonal polynomials, Duke Math. J. 5 (1939), 401–417. Google Scholar