Multi-Integral Representations for Associated Legendre and Ferrers Functions

For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order, including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind, including parameter values for which this function is identically zero.


Introduction
We have previously obtained antiderivatives and integral representations for the associated Legendre and Ferrers functions of the second kind with degree and order equal to within a sign while using analysis for fundamental solutions of the Laplace equationon Riemannian manifolds of constant curvature. For instance, we derive an antiderivative and an integral representation for the Ferrers function of the second kind with order equal to the negative degree in ([1] Theorem 1), using the d-dimensional hypersphere with d = 2, 3, 4, . . .. In ([2] Theorem 3.1), using the d-dimensional hyperboloid model of hyperbolic geometry with d = 2, 3, 4, . . ., the authors derived an antiderivative and an integral representation for the associated Legendre function of the second kind with degree and order equal to each other. In [1,2], the antiderivatives and integral representations were restricted to values of the degree and order ν such that 2ν is an integer. One of the goals of this paper is to generalize some integral representation results presented in [1,2] for the associated Legendre and Ferrers functions of the first and second kind, and to extend them, such that the degree and order are no longer subject to the above restriction. Our integral representations are consistent with the known special values for the associated Legendre and Ferrers functions of the first kind when the order is equal to the negative degree.
Another useful result we are going to use often along this work is the following.

Lemma 2.
Let n ∈ N 0 , a, x, µ ∈ C, and let f µ be a function, such that where λ µ ∈ C * . Then, the following identity holds: Proof. We are going to prove the result by induction on n. The n = 0 case is direct taking into account the Taylor expansion of f at x = a. If n = 1, then By using the Taylor expansion of f µ∓1 (x) at x = a and (13), the result follows for the n = 1 case. Assuming that the result holds for n, let us prove the identity for the n + 1 case: where we have used induction and the basic properties of integrals. Hence, the result follows.

Associated Legendre Functions of the First and Second Kind
Associated Legendre functions (and Ferrers functions) are those Gauss hypergeometric functions, which satisfy a quadratic transformation (see ( [3] Sections 15.8(iii-iv))). In the following sections, we will derive derivative, antiderivative, and integral representations for associated Legendre (and Ferrers) functions of the first and second kindswhich to the best of our knowledge have not appeared in the classical literature of these highly applicable special functions of applied and pure mathematics.
The associated Legendre function of the first kind P Starting with (14), setting µ → −µ, and applying (4), another useful hypergeometric representation for the associated Legendre function of the first kind can be obtained, namely The associated Legendre function of the second kind Q µ ν : C \ (−∞, 1] → C can be defined in terms of the Gauss hypergeometric function as ( [3] (14.3.10) and Section 14.21) where we have used (16) and (3). Furthermore algebraic expressions for Q ν+n ν for all n ∈ N are obtainable from the order recurrence relation for associated Legendre functions of the second kind (cf. ( [3] (14.10.6))) We now present some theorems that are related to the behavior of the associated Legendre function of the second kind with degree ν = − 3 2 − n ∈ {− 3 2 , − 5 2 , . . .}, n ∈ N 0 and its corresponding asymptotics as z → ∞. This will be useful in our further analysis below.
The following corollary is an interesting side-effect of the above theorem, which identifies parameter values for which the the associated Legendre function of the second kind is identically zero.
for all n ≥ k.
Proof. Simple examination of the factor multiplying the Gauss hypergeometric function in (20) produces the result.

Remark 3.
Note that the parameter values for which the associated Legendre function of the second kind is identically zero for k = 0 in Corollary 1 is clear from the special value [3] (14.5.17) .
We now give a result that produces the large argument asymptotics for the associated Legendre function of the second kind when the degree ν = − 3 2 − n, n ∈ N 0 . Equivalently, Proof. The result follows by starting with (20) and examining its leading term behavior as z → ∞.

The associated Legendre function of the second kind
We now compute some antiderivatives and integral representations for associated Legendre functions of the second kind. This also includes some nice limits and specializations.

Remark 4.
Note the following expression can be obtained by using the definition (16) and Lemma 1 for From this formula, the following antiderivative is obtained: where C is an arbitrary constant.
Proof. Taking the limit of the antiderivative (25) evaluated at the endpoints of integration using the large argument asymptotics ([3] (14.8.15)) and Lemma 3, which shows that Q . .} as well. Therefore, the integral is convergent, as indicated, which completes the proof. (24), then using induction with (1), the following order-shift derivative formula for the associated Legendre function of the second kind, namely for z ∈ C \ (−∞, 1], n ∈ N 0 , ν, µ ∈ C, holds:

Remark 7.
An antiderivative of an algebraic function (essentially in terms of reciprocal powers of the hyperbolic sine function) expressed as the associated Legendre function of the second kind with order and degree equal to each other can be obtained. This is accomplished by starting with (25) and setting µ = ν + 1, then using (18). This produces the specialized antiderivative, namely for where C is an arbitrary constant.
A straightforward consequence of the antiderivative (30) is the following integral representation for the associated Legendre function of the second kind with degree and order equal to each other.
Proof. Evaluating the antiderivative Theorem 7 at the endpoints of integration and taking advantage of ([3] (14.9.14)) Q −µ

The associated Legendre function of the first kind
An integral representation for the associated Legendre function of the first kind by applying the Whipple formulae to (27) can be obtained. However, this integral representation shifts the degree of the associated Legendre function of the first kind ν by unity instead of shifting the order by unity.
Proof. Apply the Whipple formula ([3] (14.9.16)) to the associated Legendre functions of the second kind on the left and right-hand sides of (27), followed by the application of the involution ([9] Section 2) ζ(z) := log coth z 2 and making a change of variables w = ζ/ ζ 2 − 1 completes the proof.
The following integral representation can be derived by applying Whipple's formulae to our integral representation for the associated Legendre function of the second kind. We are able to obtain an integral representation for the associated Legendre function of the first kind, which shifts the order by an integer value, similar to (28). This is achieved by deriving a corresponding derivative formula, as follows.

Remark 8.
If you divide both sides of (15) by (z 2 − 1) µ 2 and differentiate with respect to z, by using (7), then the unit increments of the parameters of the Gauss hypergeometric function can be absorbed in the order (µ) of the associated Legendre function of the first kind. Let n ∈ N 0 , z ∈ C \ (−∞, 1], ν, µ ∈ C. Then, From the above result, integral representations can be obtained through repeated integration. For instance, the single integral result is given, as follows.
Proof. In order to derive this result, after applying the fundamental theorem of calculus for some and taking advantage of ([3] (14.8.7)) this completes the proof.
The above result can be generalized by repeatedly integrating the above formula.
Proof. Repeated integration of (33) while noting (36), and using induction with (1) derives the two sum expression (53). By rewriting the associated Legendre function of the first kind on the right-hand side of (38) in terms of the Gauss hypergeometric representation (15), the finite sum term cancels the first n terms of the k sum, and rewriting the resulting expression shows it can be written in terms of a nonterminating 3 F 2 . This completes the proof.

Remark 9.
It is clear that Theorem 4 is a generalization of ([3] (14.6.7)) by considering the specialization µ = 0 in (37), which follows by using (2) and ([3] (14.9.13)) We are able to derive various expressions for the associated Legendre functions with the order equal to plus or minus the degree using this special value and the connection properties of associated Legendre functions.
An interesting definite integral follows from the behavior of the above integral representation near the singularity at z = 1. Using ([3] (14.9.15)), then After replacement of (30) and (42) in (41), we obtain Which is simply a re-evaluation of (41). From the previous identities, the following result follows.
Proof. The formula follows after a straightforward calculation starting from (44), making the change of variables w = ζ/ ζ 2 − 1 in the first integral and taking (41) into account.

Ferrers Functions of the First and Second Kind
The Ferrers functions of the first and second kinds (associated Legendre functions of the first and second kinds on-the-cut) P where we have applied the Euler transformation (4) to the single summation definition of the Ferrers function of the first kind produces the second representation for the Ferrers function of the first kind.

The Ferrers Function of the First Kind
Here, we derive interesting derivative formulae and integral representations for the Ferrers function of the first kind. First, we treat some multi-integrals of the Ferrers function of the first kind from the singularity at x = 1.
Proof. In order to derive this result, integrate (50) for n = 1 with the fundamental theorem of calculus (35) and take advantage of ([3] (14.8.1)) as x → 1 − , which completes the proof.
The above result can be generalized by repeatedly integrating the above formula.
Proof. Repeated integration of (50) while noting (52), and using induction with with (1) derives the two sum expression (53). By rewriting the Ferrers function of the first kind on the right-hand side of (54) in terms of the Gauss hypergeometric representation (46), the finite sum term cancels the first n terms of the k sum, and rewriting the resulting expression shows that it can be written in terms of a nonterminating 3 F 2 . This completes the proof.
Remark 13. It is clear that Theorem 5 is a generalization of ([3] (14.6.6)) By using the antiderivative ([3] (14.17.2)) where C is an arbitrary constant to derive some interesting integral representations for Ferrers functions of the first kind. Additionally, by using this formula to obtain a useful derivative formula for Ferrers functions of the first kind (see Remark 12).
Next, we treat some multi-integrals of the Ferrers function of the first kind from the origin. Evaluation of (57) at the endpoints of integration produces the following result. Theorem 6. Let x ∈ (−1, 1), ν, µ ∈ C. Then, Proof. Evaluating (57) at the endpoints of integration while using ( [3] (14.5.1)) completes the proof.
Proof. Repeatedly applying Theorem 6 without evaluating P µ ν (0) and then computing the Maclaurin expansion of P −µ+n ν (x)/(1 − x 2 ) (µ−n)/2 yields the first expression. Using induction evaluating P −µ+n ν (0) with (1) produces the second expression. The third expression is obtained by starting with the first expression, evaluating P −µ+n−k ν (0), shifting the sum index by n, and splitting the sum into even and odd parts.
On the other hand, by applying the antiderivative ([3] (14.17.1)) where C is an arbitrary constant to derive some interesting integral representations for Ferrers functions of the first kind. Additionally, utilizing this formula to obtain a useful derivative formula for Ferrers functions of the first kind.

Remark 14.
Differentiating the above result produces the following formula for x ∈ (−1, 1), ν, µ ∈ C, An evaluation of (63) at the endpoints of integration produces the following result.
Proof. In order to derive this result, integrate (64) for n = 1 with the fundamental theorem of calculus (35) and taking advantage of cf. (52) This completes the proof.
Proof. Repeatedly applying Theorem 8 through induction proves the result.

The Ferrers Function of the Second Kind
The Ferrers function of the second kind (associated Legendre function of the second kind on-the-cut) Q µ ν : (−1, 1) → C is defined in (47). First, we treat some multi-integrals of the Ferrers function of the second kind to the singularity at x = 1.
Proof. The Ferrers function of the second kind as x approaches the singularity at x = 1 has the following behavior ([3] (14.8.6)) as x → 1 − , µ > 0. Evaluating ( [3] (14.17.1)) while using the Ferrers function of the second kind at the endpoints of integration noting the above behavior at x ≈ 1 completes the proof.
Remark 16. Applying the fundamental theorem of calculus (35) to Lemma 4 produces the following derivative formula for x ∈ (−1, 1), ν, µ ∈ C, namely Theorem 12. Let n ∈ N 0 , x ∈ (−1, 1), ν, µ ∈ C. Then, Proof. Repeatedly applying Lemma 4 to itself using induction with (1) produces the first formula. The second formula is obtained by rewriting the finite sum as a sum from 0 to ∞ and subtracting the sum from n to ∞, and then finally utilizing (47).
Remark 17. Taking the µ → 0 limit in Theorem 12 produces the following multi-integration result for x ∈ (−1, 1), ν, µ ∈ C, namely Now, we present a similar result for the Ferrers function of the second kind with order µ instead of −µ.
. (78) Proof. The Ferrers function of the second kind as x approaches the singularity at x = 1 has the following behavior (cf. ( [3] (14.8.4))) as x → 1 − , µ > 0. Evaluating ( [3] (14.17.2)) while using the Ferrers function of the second kind at the endpoints of integration noting the above behavior at x ≈ 1 completes the proof.
Remark 18. Applying the fundamental theorem of calculus (35) to Lemma 5 produces the following derivative formula for x ∈ (−1, 1), ν, µ ∈ C, namely Theorem 13. Let n ∈ N 0 , x ∈ (−1, 1), ν, µ ∈ C. Then, Proof. Repeatedly applying Lemma 5 to itself using induction with (1) produces the first formula. The second formula is obtained by rewriting the finite sum as a sum from 0 to ∞ and subtracting the sum from n to ∞, and then finally utilizing (47).

Remark 19.
An interesting discussion is concerning whether Lemma 2 might be used in order to obtain new generalized hypergeometric representations for Theorems 12 and 13. In order to do this, one must compute the one-sided Taylor expansions of the relevant functions about the singular point x = 1 (the relevant functions are well-behaved at this singular point). This is readily possible, but it is not practical due to the fact that the behavior of the functions in question near the singularity changes in form, depending on whether µ≶0 (see (73) and (79)). The derivative terms in the Taylor series necessarily cross the µ = 0 boundary, so a simple result from this Lemma does not seem to be practical.