Bochner's theorem (orthogonal polynomials)

Let ${\displaystyle T_{2}=D_{x}^{2},T_{1}=D_{x}}$. Bochner's problem asks the following: consider the SL problem$${\displaystyle [f_{2}D_{x}^{2}+f_{1}D_{x}+f_{0}]y_{n}=\lambda _{n}y_{n}}$$For what values of ${\displaystyle f_{0},f_{1},f_{2}}$, are the eigenpairs ${\displaystyle (\lambda _{0},y_{0}),(\lambda _{1},y_{1}),\dots }$ such that ${\displaystyle y_{0}}$ is a polynomial of degree 0, ${\displaystyle y_{1}}$ is a polynomial of degree 1, etc?

Observe first that by plugging in the ${\displaystyle y_{0}=1}$ solution, we have ${\displaystyle f_{0}(x)=\lambda _{0}}$, so we may WLOG assume that ${\displaystyle \lambda _{0}=0,f_{0}=0}$. Similarly, by plugging in the ${\displaystyle y_{1}=x+a,\;y_{2}=x^{2}+ax+b}$, we find that ${\displaystyle f_{1},f_{2}}$ must be polynomials of degree at most 1, 2 respectively.

Bochner's theorem states that, up to a complex-affine transform of ${\displaystyle x}$ (that is, of form ${\displaystyle x\mapsto ax+b}$), there are only 5 families of solutions:^[2]

5 polynomial families

Polynomial solution ${\displaystyle y_{n}(x)}$	${\displaystyle f_{2}(x)}$	${\displaystyle f_{1}(x)}$	${\displaystyle \lambda _{n}}$	Condition
Jacobi polynomials ${\displaystyle P_{n}^{(\alpha ,\beta )}(x)}$	${\displaystyle 1-x^{2}}$	${\displaystyle \beta -\alpha -x(\alpha +\beta +2)}$	${\displaystyle -n(n+\alpha +\beta +1)}$	${\displaystyle \alpha ,\beta ,\alpha +\beta +1\not \in \{-1,-2,\dots \}}$
Laguerre polynomials ${\displaystyle L_{n}^{(\alpha )}(x)}$	${\displaystyle x}$	${\displaystyle 1+\alpha -x}$	${\displaystyle -n}$	${\displaystyle \alpha \not \in \{-1,-2,\dots \}}$
Hermite polynomials ${\displaystyle H_{n}(x)}$	${\displaystyle 1}$	${\displaystyle -2x}$	${\displaystyle -2n}$
Bessel polynomials ${\displaystyle y_{n}(x;a,1)}$	${\displaystyle x^{2}}$	${\displaystyle ax+1}$	${\displaystyle n(n+a-1)}$	${\displaystyle a\not \in \{0,-1,-2,\dots \}}$
Monomials ${\displaystyle x^{n}}$	${\displaystyle x^{2}}$	${\displaystyle x}$	${\displaystyle n^{2}}$

Proof sketch: By the previous observation, there are only 5 parameters in total that characterize the prob/em:$${\displaystyle (ax^{2}+bx+c)y''+(dx+e)y'=\lambda y}$$Setting ${\displaystyle a=1}$, then up to a complex affine transform, it reduces to the form$${\displaystyle (1-x^{2})y''+(dx+e)y'=\lambda y}$$This is the form of the Jacobi differential equation, and has polynomial solutions precisely when there exists ${\displaystyle \alpha ,\beta }$ such that ${\displaystyle dx+e=\beta -\alpha -x(\alpha +\beta +2)}$. These are the Jacobi polynomials. The other cases are, up to affine scaling, the various limits of the ${\displaystyle a=1}$ case. The solution families are then obtained by taking the respective limits of the Jacobi polynomials. The conditions on the parameters are necessary to prevent the leading coefficient from going zero.^[3]

The original proof by Bochner directly considered the 3 possible cases:

• ${\displaystyle a=1}$
• ${\displaystyle a=0,b=1}$
• ${\displaystyle a=b=0,c=1}$

and for each, performed a complex-affine transform of the variable and solved the corresponding equation.^[1][2]

The Bochner's theorem allows many extensions, by relaxing various conditions on the setup of Bochner's problem. Other extensions are reviewed in ^[4] and.^[5]

If instead of complex-affine transforms, we only permit real-affine transforms, then there are 2 more families: twisted Hermite, and twisted Jacobi:^[2]

2 extra polynomial families

Polynomial solution ${\displaystyle y_{n}(x)}$	${\displaystyle f_{2}(x)}$	${\displaystyle f_{1}(x)}$	${\displaystyle \lambda _{n}}$	Condition
twisted Jacobi polynomials ${\displaystyle {\check {P}}_{n}^{(\alpha ,\beta )}(x)}$	${\displaystyle 1+x^{2}}$	${\displaystyle i(\beta -\alpha )-x(\alpha +\beta +2)}$	${\displaystyle n(n+\alpha +\beta +1)}$	${\displaystyle \alpha +\beta +1\not \in \{-1,-2,\dots \}}$
twisted Hermite polynomials ${\displaystyle {\check {H}}_{n}(x)}$	${\displaystyle 1}$	${\displaystyle 2x}$	${\displaystyle 2n}$

The 2 extra cases are orthogonal, but not positive-definite. The twisted Hermite satisfies the following complex-orthogonality:$${\displaystyle \int _{-i\infty }^{i\infty }{\check {H}}_{m}(x){\check {H}}_{n}(x)\exp \left(x^{2}\right)dx=(-1)^{n}{\sqrt {\pi }}n!2^{-n}i\delta _{mn}\quad (m{\text{ and }}n\geq 0)}$$They are orthogonal (but not positive-definite) with respect to the real weight function$${\displaystyle w(x)={\begin{cases}0&{\text{ if }}x\leq 0\\e^{x^{2}}\int _{0}^{x}e^{-t^{2}}g(t)dt&{\text{ if }}x>0\end{cases}}}$$where ${\displaystyle g}$ is any real function supported on ${\displaystyle [0,\infty )}$, that satisfies ${\displaystyle \int _{0}^{\infty }e^{-x^{2}}g(x)dx=0}$, and has all moments zero.

The twisted Jacobi satisfies the following complex-orthogonality:$${\displaystyle {\begin{aligned}&{\begin{aligned}&\left\langle (1-x)_{+}^{\alpha }(1+x)_{+}^{\beta },{\check {P}}_{m}^{(\alpha ,\beta )}(ix){\check {P}}_{n}^{(\alpha ,\beta )}(ix)\right\rangle \\&={\frac {(-1)^{n}2^{2n+\alpha +\beta +1}\Gamma (n+\alpha +1)\Gamma (n+\beta +1)\Gamma (n+\alpha +\beta +1)n!}{\Gamma (2n+\alpha +\beta +1)\Gamma (2n+\alpha +\beta +2)}}\delta _{mn}\end{aligned}}\\&(m{\text{ and }}n\geq 0)\end{aligned}}}$$where the average is taken over the function ${\displaystyle \left(1+x^{2}\right)^{\frac {2-d}{2}}e^{i(\beta -\alpha )\arctan x}}$.

They are orthogonal (but not positive-definite) with respect to the real weight function$${\displaystyle w(x)=e^{f(x)}\left[c+\int _{0}^{x}e^{-f(t)}\left(1+t^{2}\right)^{-1}g(t)dt\right]}$$where ${\displaystyle g}$ is any real function supported on ${\displaystyle [0,\infty )}$, that satisfies ${\displaystyle \int _{0}^{\infty }e^{-f(t)}\left(1+t^{2}\right)^{-1}g(t)dt=0}$, and has all moments zero.

The original Bochner's theorem assumes that there exists one polynomial solution per degree. If we relax this assumption, then we obtain the exceptional polynomial families. In an exception polynomial family, some degrees may not correspond to any polynomial. Such a family does not make up a complete basis for the function space, because certain polynomial functions cannot be expressed as a linear combination of its elements, but such a family may still have certain applications.^[6][7]