Why are there two different recurrences for Gegenbauer polynomials?

Question

As I mentioned previously, I've been reading up on Gegenbauer polynomials in preparation for a blog post on the kissing number problem—specifically, the Delsarte method.

To make a long story short, the method involves expressing a particular function as a non-negative linear sum of Gegenbauer polynomials. In various publications on this topic (see, for example, Musin, "The Kissing Number in Four Dimensions" in the July 2008 Annals of Mathematics), these polynomials have the following recurrence:

$$ G^{(n)}_0(t) = 1 $$ $$ G^{(n)}_1(t) = t $$ $$ G^{(n)}_k(t) = \frac{(2k+n-4)tG^{(n)}_{k-1}(t)-(k-1)G^{(n)}_{k-2}(t)}{k+n-3} $$

However, in the Wikipedia and Wolfram MathWorld plot summaries for Gegenbauer polynomials, the recurrences are different (and not just up to a constant factor). In both, converting to consistent symbols, we have

$$ G^{(n)}_0(t) = 1 $$ $$ G^{(n)}_1(t) = 2nt $$ $$ G^{(n)}_k(t) = \frac{2t(k+n-1)G^{(n)}_{k-1}(t)-(k+2n-2)G^{(n)}_{k-2}(t)}{k} $$

Both definitions are normalized to $G^{(n)}_0(t) = 1$. What accounts for the difference between the two?

Answer 1

I'll use $G$ for the second definition and $\widetilde G$ for the first definition. $G$ has the generating function $(1 - 2 x t + t^2)^{-n}$: $$(1 - 2 x t + t^2)^{-n} = \sum_{k \geq 0} G_k^{(n)}(x) \,t^k.$$ $\widetilde G$ is constructed by first taking $C$ with the gf $(1 - 2 x t + t^2)^{1 -n/2}$ and then normalizing by $C(1)$: $$(1 - 2 x t + t^2)^{1 - n/2} = \sum_{k \geq 0} C_k^{(n)}(x) \,t^k, \\ \widetilde G_k^{(n)}(x) = \frac {C_k^{(n)}(x)} {C_k^{(n)}(1)}.$$ Therefore $$\widetilde G_k^{(n)}(x) = \frac {(-1)^k} {\binom {2 - n} k } G_k^{(n/2 - 1)}(x).$$

Answer 2

Here are some aspects which might help to clarify the situation. At first I'd like to give a few definitions of Gegenbauer polynomials from relevant sources. This way we can get an impression what we typically might expect.

Higher Transcendental Functions, Vol I by A. Erdelyi and H. Bateman (author):

• (3.15.1. Gegenbauer polynomials)

  Gegenbauer's polynomial $C_n^{\nu}(z)$ for integral value $n$ is defined to be the coefficient of $h^n$ in the expansion of $(1-2hz+h^2)^{-\nu}$ in powers of $h$;

\begin{align*} (1-2hz+h^2)^{-\nu}=\sum_{n=0}^\infty C_n^{\nu}(z)h^n\qquad\qquad|h|<|z\pm(z^2-1)^{1/2}| \end{align*}

Handbook of Mathematical Functions by M. Abramowitz, I.A. Stegun:

• (22.9.3. Generating Functions)

\begin{align*} (1-2xz+z^2)^{-\alpha}=\sum_{n=0}^\infty C_n^{(\alpha)}(x)z^n\qquad\qquad |z|<1,\alpha\ne 0 \end{align*}

NIST/DLMF

• (18.12.4. Ultraspherical)

\begin{align*} (1-2xz+z^2)^{-\alpha}=\sum_{n=0}^\infty C_n^{(\alpha)}(x)z^n\qquad\qquad |z|<1 \end{align*}

Special Functions, Encyclopedia of mathematics and its applications 71 by G.E. Andrews, R. Askey and R. Roy:

• (6.4 Generating Functions for Jacobi Polynomials)

  It is reasonable to define polynomials

\begin{align*} C_n^{\lambda}(x):=\frac{(2\lambda)_n}{(\lambda+(1/2))_n}P_n^{(\lambda-(1/2),\lambda-(1/2))}(x)\tag{1} \end{align*}

$\qquad$with the generating function

\begin{align*} (1-2xr+r^2)^{-\lambda}=\sum_{n=0}^\infty C_n^{\lambda}(x)r^n.\tag{2} \end{align*}

Note the exponent of $(1-2xr+r^2)^{\color{blue}{-\lambda}}$ and the upper index in $C_n^{\color{blue}{\lambda}}(x)$ in (2) are coupled by a multiplicative factor $-1$ and this notational convention is used in all these citations. The connection with the Jacobi polynomials is given in (1).

We take a look at OPs cited paper The Kissing Number in Four Dimensions by O.R. Musin, check a few statements and a cited reference we are interested in.

• (3.B The Gegenbauer polynomials)

  ... Let us recall definitions of Gegenbauer polynomials $C_k^{(n)}(t)$, which are defined by the expansion

\begin{align*} (1-2rt+r^2)^{(2-n)/2}=\sum_{k=0}^\infty r^kC_k^{(n)}(t)\tag{3} \end{align*}

This definition looks somewhat peculiar, since the upper index $n$ of $C_k^{(\color{blue}{n})}$ is not coupled by a factor $-1$ with the exponent of $(1-2rt+r^2)^{\color{blue}{(2-n)/2}}$ of the generating function. A few lines above (3.B) the author states

... Schoenberg [29] extended this property to Gegenbauer polynomials $G_k^{(n)}$. He proved: The matrix $\left(G_k^{(n)}\left(\cos \phi_{i,j}\right)\right)$ is positive semidefinite for any finite $X\subseteq \mathbf{S}^{n-1}$.

The reference [29] addresses the paper Positive definite functions on spheres by I.J. Schoenberg. It is revealing to check the definition Schoenberg used for Gegenbauer (resp. ultraspherical) polynomials.

• (Schoenberg [29], section 1)

  ... Let $P_n^{(\lambda)}(\cos t)$ be the ultraspherical polynomials defined by the expansion

\begin{align*} (1-2r\cos t+r^2)^{-\lambda}=\sum_{n=0}^\infty r^nP_n^{(\lambda)}(\cos t),\qquad\qquad (\lambda>0).\tag{4} \end{align*}

Note in definition (4) the parameter $\lambda$ is used in accordance with the citations above. A few lines later Schoenberg gives a series expansion of a function $g(t)$ in terms of Gegenbauer polynomials:

• (Schoenberg [29], section 1)

  ... The most general element of $\mathcal{P}(S_m)$ is

\begin{align*} g(t)=\sum_{n=0}^\infty P_n^{(\lambda)}(\cos t),\qquad\qquad(\lambda=\frac{1}{2}(m-1)),\tag{5} \end{align*}

Conclusion: Comparing Schoenberg's usage of the parameter $\lambda$ in (5) and (4) with Musin's parameter setting in (3) indicates he could have had some short-cut notation in mind. Regrettably this implies that recurrence relation, differential equations, etc. have a different form than usually expected. The differences in the recurrence relation do not occur, if Musin would have used (3') instead

\begin{align*} (1-2rt+r^2)^{-\lambda}=\sum_{k=0}^\infty r^kC_k^{(\lambda)}(t)\qquad\qquad (\lambda=(n-2)/2)\tag{3'} \end{align*}