# Gegenbauer's addition theorem for Jacobi polynomials

I have the following identity,

$$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, P_n(z) = 2 \, j_n(x) \, j_n(y) \;,$$

where $$x, y > 0$$, $$P_n$$ is a Legendre polynomial, and $$j_n$$ is a spherical Bessel function. This follows from Gegenbauer's addition theorem (NIST 10.60.2),

$$j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) = \sum_{l \ge 0} (2l+1) \, j_l(x) \, j_l(y) \, P_l(z) \;.$$

I am interested in the generalisation of the above to general Jacobi polynomials $$P^{(\alpha, \beta)}_n$$, $$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, (1-z)^\alpha (1+z)^\beta \, P^{(\alpha,\beta)}_n(z) \;,$$ with $$\alpha, \beta > -1$$ and $$x, y > 0$$.

We have the following connection formula for Jacobi and Legendre polynomials, $$P_l(z) = \sum_{k=0}^l \tfrac{\Gamma(k+\alpha+\beta+1) \, \Gamma(k+l+1)}{\Gamma(k+1) \, \Gamma(l-k+1) \, \Gamma(2k+\alpha+\beta+1)} \, _3F_2\bigl(\genfrac..{0pt}{1}{k-l, k+l+1, k+\alpha+1}{k+1, 2k+\alpha+\beta+2}; 1\bigr) \, P^{(\alpha,\beta)}_k(z) \;,$$ with $$_3F_2$$ the hypergeometric function. With this, we can rewrite the addition theorem for general Jacobi polynomials, $$j_0\bigl(\sqrt{x^2 + y^2 - 2xyz}\bigr) = \sum_{l \ge 0} R^{(\alpha,\beta)}_l(x, y) \, P^{(\alpha,\beta)}_l(z) \;,$$ using the new coefficient functions $$R_l^{(\alpha,\beta)}(x, y) = \sum_{k \ge l} (2k + 1) \, j_k(x) \, j_k(y) \, \tfrac{\Gamma(l+\alpha+\beta+1) \, \Gamma(k+l+1)}{\Gamma(l+1) \, \Gamma(k-l+1) \, \Gamma(2l+\alpha+\beta+1)} \, _3F_2\bigl(\genfrac..{0pt}{1}{l-k, l+k+1, l+\alpha+1}{l+1, 2l+\alpha+\beta+2}; 1\bigr) \;.$$

Q: Can the functions $$R_l^{(\alpha,\beta)}(x, y)$$ be simplified, even in special cases? Are there other cases besides $$\alpha = \beta = 0$$ where the result factors as $$f(x)\,g(y)$$?

One special case might be $$\beta = 0$$, where the hypergeometric function can be replaced using Saalschütz's theorem.

(This question was updated with results from a deleted answer.)

• for reference, the second integral evaluates for $n=\alpha=\beta=1$ to $\frac{8}{x^4 y^4} \sin y \left(x^3 y^2 \cos x-3 x^3 \cos x-7 x^2 y^2 \sin x+18 x^2 \sin x+18 y^2 \sin x-18 x y^2 \cos x-45 \sin x+45 x \cos x\right)+$ $\frac{8}{x^4 y^3} \cos y \left(3 x^3 \cos x+x^2 y^2 \sin x-18 x^2 \sin x-3 y^2 \sin x+3 x y^2 \cos x+45 \sin x-45 x \cos x\right)$. Dec 17 '19 at 20:01
• This is direct consequence of partial wave expansion of $\frac{\sin R}{R}$, where $R=\sqrt{x^2+y^2-2xr\cos\theta}$. I think it might be found in Morse-Feshbach textbook. But I doubt it can be generalized for Jacobi polynomials instead of Legendre polynomials.
– Nemo
Dec 17 '19 at 20:02
• A deleted answer showed the first result straightforwardly via the addition theorem DLMF (10.60.2) / (10.23.8). The first result can of course be naturally'' generalised by expressing one set of polynomials in terms of the other, but that does not lead to a final result which factorises in the same way.
– user114668
Dec 18 '19 at 11:19

Too long for a comment: Since the Legendre polynomials define the zonal harmonics on the sphere, a similar formula should hold at least for the Jacobi polynomials which correspond to harmonics for the other two-point homogeneous spaces. The spherical Bessel function may not appear in this case, but instead some other function. If you are particulary interested in an equation which is exactly in the form as the spherical $$d=3$$ case, the below will likely not be very relevant.

For compact two-point homogeneous spaces generally, there exists a decomposition of $$L^2(\Omega)$$, the space of complex-valued square-integrable functions on $$\Omega$$, into irreducible representations: $$L^2(\Omega)=\bigoplus\limits_{n\geq 0}V_n ,$$ where spaces $$V_n$$ are finite-dimensional and invariant under the isometries of $$\Omega$$. Moreover, they can be chosen as the eigenspaces of the Laplace--Beltrami operator on $$\Omega$$ corresponding to the $$n$$-th eigenvalue in the increasing order. Let $$Y_{n,k}$$, $$k =1,\ldots,\dim V_n$$, be an orthonormal basis in $$V_n$$. Because of the invariance of $$V_n$$ and due to the two-point homogeneity of $$\Omega$$, the reproducing kernel for $$V_n$$ only depends on the distance $$\vartheta(x,y)$$ between points . Furthermore, as a function of $$\tau(x,y) := \cos \vartheta(x,y)$$ the reproducing kernel is a polynomial $$C_n$$ of degree $$n$$, which satisfies $$C_n(\tau(x,y))= \frac1{\dim V_n} \sum\limits_{k=1}^{\dim V_n} Y_{n,k}(x) \overline{Y_{n,k}(y)}.$$

The formula above is known as the addition formula on these spaces.

The Jacobi polynomials $$C_n$$ given by the addition formula satisfy $$C_n(1) = 1$$ and are orthogonal with respect to the probability measure $$d\nu^{(\alpha, \beta)} = \frac1{\gamma_{\alpha,\beta}} (1-t)^{\alpha} (1+t)^{\beta} dt,$$ where $$\alpha = (d-1) \dim_{\mathbb R} (\mathbb F)/{2}-1$$ and $$\beta=\alpha, \text{ if } \Omega = \mathbb{S}^{d-1} \\$$ $$\beta={\dim_{\mathbb{R}}(\mathbb{F})}/{2} -1, \text{ if } \Omega = \mathbb{FP}^{d-1},$$ and the normalization factor is given by $$\gamma_{\alpha,\beta} = 2^{\alpha +\beta +1}B(\alpha+1, \beta+1),$$ where $$B$$ is the beta function.

• Thank you for your answer! I am in fact interested in finding cases with other functions on the rhs. of the addition theorem. If you had a reference for the harmonics of other two-point homogeneous spaces, that could be immensely helpful.
– user114668
Dec 19 '19 at 16:46
• Wang classified all the compact two-point homogeneous spaces: H.-C. Wang.Two-Point Homogeneous Spaces. Ann. of Math.55(1952), 177–191. MR0047345, and Gangolli's paper: Positive definite kernels on homogeneous spaces and certain stochastic processes related to L ́evy’s Brownian motion of several parameters. Ann. Inst. H. Poincar ́e Sect. B (N.S.)3(1967), 121–226. MR0215331, might also be relevant. There are also more recent papers which deal with expansions into harmonics, such as the following: M. M. Skriganov. Point distribution in compact metric spaces, III. Two-point homogeneous spaces. Dec 19 '19 at 17:00
• Levenshtein's paper does this well too, although it is not primarily concerned with the expansions, but rather applications. V. I. Levenshtein. Designs as maximum codes in polynomial metric spaces. Acta Appl. Math.29(1992),1–82. MR1192833 Dec 19 '19 at 17:10