Differential equations for generalized Laguerre and Jacobi polynomials

In [9] Tom H. Koornwinder introduced the polynomials \(\{P_n^{\alpha,\beta,M,N}(x)\}_{n=0}^{\infty}\) which are orthogonal on the interval \([-1,1]\) with respect to the weight function \[\frac{\Gamma(\alpha+\beta+2)}{2^{\alpha+\beta+1}\Gamma(\alpha+1)\Gamma(\beta+1)} (1-x)^{\alpha}(1+x)^{\beta}+M\delta(x+1)+N\delta(x-1),\] where \(\alpha>-1\), \(\beta>-1\), \(M\geq0\) and \(N\geq0\). We call these polynomials the generalized Jacobi polynomials, but sometimes they are also referred to as the Jacobi-type polynomials. As a limit case he also found the generalized Laguerre (or Laguerre-type) polynomials \(\{L_n^{\alpha,M}(x)\}_{n=0}^{\infty}\) which are orthogonal on the interval \([0,\infty)\) with respect to the weight function \[\frac{1}{\Gamma(\alpha+1)}x^{\alpha}e^{-x}+M\delta(x),\] where \(\alpha>-1\) and \(M\geq0\). These generalized Jacobi polynomials and generalized Laguerre polynomials are related by the limit \[L_n^{\alpha,M}(x)=\lim\limits_{\beta\rightarrow\infty} P_n^{\alpha,\beta,0,M}\left(1-\frac{2x}{\beta}\right).\] In [3] we proved that for \(M\geq0\) the generalized Laguerre polynomials satisfy a unique differential equation of the form \[M\sum_{i=0}^{\infty}a_i(x)y^{(i)}(x)+xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0,\] where \(\{a_i(x)\}_{i=0}^{\infty}\) are continuous functions on the real line and \(\{a_i(x)\}_{i=1}^{\infty}\) are independent of the degree \(n\). In [1] Herman Bavinck found a new method to obtain the main result of [3]. This inversion method was found in a similar way as was done in [2] in the case of generalizations of the Charlier polynomials. See also [5] for more details. In [7] we used this inversion method to find all differential equations of the form \[\begin{array}{l}M\sum\limits_{i=0}^{\infty}a_i(x)y^{(i)}(x)+N\sum\limits_{i=0}^{\infty}b_i(x)y^{(i)}(x)\\ {}\hspace{10mm}{}+MN\sum\limits_{i=0}^{\infty}c_i(x)y^{(i)}(x)+xy''(x)+(\alpha+1-x)y'(x)+ny(x)=0,\end{array}\] where the coefficients \(\{a_i(x)\}_{i=1}^{\infty}\), \(\{b_i(x)\}_{i=1}^{\infty}\) and \(\{c_i(x)\}_{i=1}^{\infty}\) are independent of \(n\) and the coefficients \(a_0(x)\), \(b_0(x)\) and \(c_0(x)\) are independent of \(x\), satisfied by the Sobolev-type Laguerre polynomials \(\{L_n^{\alpha,M,N}(x)\}_{n=0}^{\infty}\) which are orthogonal with respect to the inner product \[\langle f, g\rangle=\frac{1}{\Gamma(\alpha+1)}\int_0^{\infty}x^{\alpha}e^{-x}f(x)g(x)\,dx +Mf(0)g(0)+Nf'(0)g'(0),\] where \(\alpha>-1\), \(M\geq0\) and \(N\geq0\). These Sobolev-type Laguerre polynomials \(\{L_n^{\alpha,M,N}(x)\}_{n=0}^{\infty}\) are generalizations of the generalized Laguerre polynomials \(\{L_n^{\alpha,M}(x)\}_{n=0}^{\infty}\). In fact we have \[L_n^{\alpha,M,0}(x)=L_n^{\alpha,M}(x).\] In [6] we used the inversion formula found in [5] to find differential equations of the form \[\begin{array}{lr}M\sum\limits_{i=0}^{\infty}a_i(x)y^{(i)}(x)+N\sum\limits_{i=0}^{\infty}b_i(x)y^{(i)}(x) +MN\sum\limits_{i=0}^{\infty}c_i(x)y^{(i)}(x)+(1-x^2)y''(x) &\\ {}\hspace{1cm}{}+[\beta-\alpha-(\alpha+\beta+2)x]y'(x)+n(n+\alpha+\beta+1)y(x)=0, & (1)\end{array}\] where the coefficients \(\{a_i(x)\}_{i=1}^{\infty}\), \(\{b_i(x)\}_{i=1}^{\infty}\) and \(\{c_i(x)\}_{i=1}^{\infty}\) are independent of \(n\) and the coefficients \(a_0(x)\), \(b_0(x)\) and \(c_0(x)\) are independent of \(x\), satisfied by the generalized Jacobi polynomials \(\{P_n^{\alpha,\beta,M,N}(x)\}_{n=0}^{\infty}\). We gave explicit representations for the coefficients \(\{a_i(x)\}_{i=0}^{\infty}\), \(\{b_i(x)\}_{i=0}^{\infty}\) and \(\{c_i(x)\}_{i=0}^{\infty}\) and we showed that this differential equation is uniquely determined. For \(M^2+N^2>0\) the order of this differential equation is infinite, except for \(\alpha\in\{0,1,2,\ldots\}\) or \(\beta\in\{0,1,2,\ldots\}\). Moreover, the order equals \[\left\{\begin{array}{ll} 2\beta+4 & \textrm{ if } M>0, N=0 \textrm{ and } \beta\in\{0,1,2,\ldots\}\\[2.5mm] 2\alpha+4 & \textrm{ if } M=0, N>0 \textrm{ and } \alpha\in\{0,1,2,\ldots\}\\[2.5mm] 2\alpha+2\beta+6 & \textrm{ if } M>0, N>0 \textrm{ and } \alpha,\beta\in\{0,1,2,\ldots\}.\end{array}\right.\] For \(\alpha=\beta=0\), \(M>0\) and \(N>0\) the generalized Jacobi polynomials reduce to the Krall polynomials studied by Lance L. Littlejohn in [13]. These Krall polynomials are generalizations of the Legendre type polynomials (\(\alpha=\beta=0\) and \(N=M>0\)) found by H.L. Krall in [11] and [12]. See also [10]. In [13] it is shown that the Krall polynomials satisfy a sixth order differential equation of the form (1). For \(\alpha>-1\), \(\beta=0\), \(M>0\) and \(N=0\) or for \(\alpha=0\), \(\beta>-1\), \(M=0\) and \(N>0\) the generalized Jacobi polynomials reduce to the Jacobi type polynomials which satisfy a fourth order differential equation of the form (1) ; see also [10], [11] and [12].

We emphasize that the case \(\beta=\alpha\) and \(N=M\) is special in the sense that we can also find differential equations of the form \[\begin{array}{lr}M\sum\limits_{i=0}^{\infty}d_i(x)y^{(i)}(x)+(1-x^2)y''(x)-2(\alpha+1)xy'(x)+n(n+2\alpha+1)y(x)=0, & (2)\end{array}\] where the coefficients \(\{d_i(x)\}_{i=1}^{\infty}\) are independent of \(n\) and \(d_0(x)\) is independent of \(x\), satisfied by the symmetric generalized ultraspherical polynomials \(\{P_n^{\alpha,\alpha,M,M}(x)\}_{n=0}^{\infty}\). The Legendre type polynomials for instance satisfy a fourth order differential equation of the form (2). See [10], [11] and [12]. In [8] we found all differential equations of the form (2) satisfied by the polynomials \(\{P_n^{\alpha,\alpha,M,M}(x)\}_{n=0}^{\infty}\) for \(\alpha>-1\) and \(M\geq0\). In [4] we applied the special case \(\beta=\alpha\) of the Jacobi inversion formula to solve the systems of equations obtained in [8].

References

[1]

H. Bavinck : A direct approach to Koekoek's differential equation for generalized Laguerre polynomials. Acta Mathematica Hungarica 66, 1995, 247-253.

[2]

H. Bavinck and R. Koekoek : On a difference equation for generalizations of Charlier polynomials. Journal of Approximation Theory 81, 1995, 195-206.

[3]

J. Koekoek and R. Koekoek : On a differential equation for Koornwinder's generalized Laguerre polynomials. Proceedings of the American Mathematical Society 112, 1991, 1045-1054.

[4]

J. Koekoek and R. Koekoek : Finding differential equations for symmetric generalized ultraspherical polynomials by using inversion methods. Proceedings of the International Workshop on Orthogonal Polynomials in Mathematical Physics (Leganés, 1996), Universidad Carlos III de Madrid, Leganés, 1997, 103-111.

[5]

J. Koekoek and R. Koekoek : The Jacobi inversion formula. Complex Variables 39, 1999, 1-18.

[6]

J. Koekoek and R. Koekoek : Differential equations for generalized Jacobi polynomials. Journal of Computational and Applied Mathematics 126, 2000, 1-31.

[7]

J. Koekoek, R. Koekoek and H. Bavinck : On differential equations for Sobolev-type Laguerre polynomials. Transactions of the American Mathematical Society 350, 1998, 347-393.

[8]

R. Koekoek : Differential equations for symmetric generalized ultraspherical polynomials. Transactions of the American Mathematical Society 345, 1994, 47-72.

[9]

T.H. Koornwinder : Orthogonal polynomials with weight function (1-x)^a(1+x)^b +Md(x+1)+Nd(x-1). Canadian Mathematical Bulletin 27(2), 1984, 205-214.

[10]

A.M. Krall : Orthogonal polynomials satisfying fourth order differential equations. Proceedings of the Royal Society of Edinburgh 87A, 1981, 271-288.

[11]

H.L. Krall : Certain differential equations for Tchebycheff polynomials. Duke Mathematical Journal 4, 1938, 705-718.

[12]

H.L. Krall : On orthogonal polynomials satisfying a certain fourth order differential equation. The Pennsylvania State College Studies, No. 6, 1940.

[13]

L.L. Littlejohn : The Krall polynomials : A new class of orthogonal polynomials. Quaestiones Mathematicae 5, 1982, 255-265.

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Last modified on January 2, 2013