Symmetry, Integrability and Geometry: Methods and Applications Definite Integrals using Orthogonality and Integral Transforms ⋆

We obtain definite integrals for products of associated Legendre functions with Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind using orthogonality and integral transforms.


Introduction
In [3] and [6] (see also [4]), we present some definite integral and infinite series addition theorems which arise from expanding fundamental solutions of elliptic equations on R d in axisymmetric coordinate systems which separate Laplace's equation. We utilize orthogonality and integral transforms to obtain new definite integrals from some of these addition theorems.
Proof . By applying Theorem 1 to the function F : (0, ∞) → C defined by using (5), we obtain the desired result.
Now, we give another example of how an integral expansion for a fundamental solution of Laplace's equation on R 3 in parabolic coordinates can be used to prove a new definite integral.
3 Def inite integrals from orthogonality relations 3.1 Degree orthogonality for associated Legendre functions with integer degree and order We take advantage of the degree orthogonality relation for the Ferrers function of the first kind with integer degree and order, namely (cf. [7, (7.112.1)]) π 0 P m n (cos θ)P m n (cos θ) sin θdθ = 2 2n + 1 where m, n, n ∈ N 0 , and m ≤ n, m ≤ n . We are using the associated Legendre function of the first kind (on-the-cut), P µ ν : (−1, 1) → C, for ν, µ ∈ C, the Ferrers function of the first kind, which is defined in [12, (14.3.1)].

Order orthogonality for associated Legendre functions with integer degree and order
In this subsection we take advantage of the order orthogonality relation for the Ferrers function of the first kind with integer degree and order (cf. [12, (14.17.8)]) π 0 P m n (cos θ)P m n (cos θ) with m ≥ 1.
Proof . We start with the addition theorem for spherical harmonics (cf. [12, (14.18.1)]), namely P n (cos γ) = n m=−n (n − m)! (n + m)! P m n (cos θ)P m n (cos θ )e im(φ−φ ) , where P n : C → C, for n ∈ N 0 , is the Legendre polynomial which can be defined in terms of the terminating Gauss hypergeometric series (see for instance [12,Chapters 15,18]) as follows We then take advantage of the order orthogonality relation for the Ferrers functions of the first kind with integer degree and order. If we multiply both sides of (28) by (sin θ) −1 P m n (cos θ) and integrate over θ ∈ (0, π), by using (27) we obtain the desired result.
Theorem 9, originating from (28), is the only example of a definite integral that we could find using the order orthogonality relation for the Ferrers functions of the first kind (27). Therefore we highly suspect that this result is previously known, and include it mainly for completeness sake. It would however be very interesting to find another example using this orthogonality relation.

Orthogonality for Chebyshev polynomials of the f irst kind
Here we take advantage of orthogonality from Chebyshev polynomials of the first kind (cf. [12, § 18.3]) π 0 T m (cos θ)T n (cos θ)dθ = π n δ m,n , where T n : C → C, for n ∈ N 0 , is the Chebyshev polynomial of the first kind which can be defined in terms of the terminating Gauss hypergeometric series (see [12, Chapter 18]) T n (z) = 2 F 1 −n, n 1 2 The Chebyshev polynomials of the first kind satisfy the identity [12, (18.5.1)] T n (cos θ) = cos(nθ).
Proof . We start with toroidal coordinates on R 3 , namely where a > 0, σ ∈ (0, ∞), ψ, φ ∈ [0, 2π). The reciprocal distance between two points x, x ∈ R 3 is given algebraically by where (σ , ψ , φ ) are the toroidal coordinates corresponding to the point x . Using Heine's reciprocal square root identity (see for instance [5, (3.11) where z > 1 and x ∈ [−1, 1], we can obtain a Fourier cosine series representation for the reciprocal distance between two points in toroidal coordinates on R 3 , namely where χ > 1 is given by (31). We can further expand the associated Legendre function of the second kind using the following addition theorem (cf. [7, (8.795.