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Open AccessArticle

Multi-Integral Representations for Associated Legendre and Ferrers Functions

1
Applied and Computational Mathematics Division, National Institute of Standards and Technology, Mission Viejo, CA 92694, USA
2
Departamento de Física y Matemáticas, Universidad de Alcalá, c.p. 28871 Alcalá de Henares, Spain
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2020, 12(10), 1598; https://doi.org/10.3390/sym12101598
Received: 29 August 2020 / Revised: 16 September 2020 / Accepted: 17 September 2020 / Published: 25 September 2020
(This article belongs to the Special Issue Symmetry in Special Functions and Orthogonal Polynomials)

Abstract

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.
Keywords: associated legendre functions; ferrers functions; integral representations; gauss hypergeometric function associated legendre functions; ferrers functions; integral representations; gauss hypergeometric function

1. 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 2 ν 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.
The multi-integrals presented in this paper (Theorems 3–5, 7 and 11) generalize multi-integrals for arbitrary order ( μ ) , which have appeared previously in the literature (see, for instance, ([3] Section 14.6(ii)), ([4] 8.14.17–18) and the earliest appearance we have found ([5] p. 149)). The multi-integrals for the Ferrers function of the second kind (Theorems 12, 13, 15 and 17) also produce generalized results for arbitrary order ( μ ) . In fact, the specialization of these multi-integrals for μ = 0 has not appeared in the literature.
Applications of the work that is contained in this manuscript include any of the many different areas in which Legendre and Ferrers functions arise, which include a very large number of disciplines. The associated Legendre and Ferrers functions treated in this paper, e.g., Q ν μ ( cosh r ) / sinh μ r , appear as fundamental solutions of the Laplace and Helmholtz equation on Riemannian manifolds of constant curvature (see e.g., ([6] Section 3.3)). Associate Legendre and Ferrers functions appear in any place where harmonic analysis needs to be performed on the surface of a sphere or on an oblate or prolate spheroid. These are analogs of the 1 / r potential in Euclidean space for Riemannian spaces of constant curvature. Therefore, results, such as we derive below, will be important when studying global analysis of these fundamental solutions for higher powers of the Laplacian or Helmholtz operators. These multi-integration results provide an algorithm for computing fundamental solutions of much larger powers of the Laplace–Beltrami operator on these spaces. A survey of applications of associated Legendre and Ferrers functions is given in ([3] Section 14.30-31). This points to harmonic analysis on the surface of spheres, oblate and prolate spheroids, and circular toroids. Other applications include the Mehler–Fock transforms, high frequency atomic and molecular scattering, quantum direct and exchange Coulomb interaction, Newtonian gravity, etc. Additionally, the special cases treated in this paper, many of which are also not well-known, provide for a beautiful illumination of this classical subject.
The critical aspects that allows for proofs of the results contained in this paper is that fact that we are able to obtain differentiation/integration properties of associated Legendre and Ferrers functions, such that the order ( μ ) of these functions are either raised or lowered by integral amounts. Furthermore, the differentiation/integration no not affect the degree ( ν ) of these functions at all. In fact, these differentiation/integration properties (Remarks 5, 8, 12, 14, 16, 18, 22 and 24) are not well-known in the literature for these functions.
Note that in the process of our derivations, we also obtained some nice results for the associated Legendre function of the second kind with degree ν = 3 2 n . This included the full Gauss hypergeometric dependence and large argument asymptotics, which is crucial for establishing Theorem 3. We are also able to find parameter values for which the function is identically zero that occur when μ = ± ( 1 2 + k ) , n , k N 0 , n k (see Corollary 1).

2. Preliminaries

Throughout this paper, we adopt the following set notations: N 0 { 0 } N = { 0 , 1 , 2 , 3 , } , and we use the set C , which represents the complex numbers. As is the common convention for associated Legendre functions ([4] (8.1.1)), for any expression of the form ( z 2 1 ) α , read this as ( z 2 1 ) α ( z + 1 ) α ( z 1 ) α , for any fixed α C and z C \ ( , 1 ] . In this paper, we will use the Gauss hypergeometric function 2 F 1 , which can be defined in terms of the following infinite series, as ([7] (2.1.5))
2 F 1 a , b c ; z n = 0 ( a ) n ( b ) n ( c ) n z n n ! ,
where c N 0 , and, elsewhere, on z C \ ( 1 , ) by analytic continuation; where the Pochhammer symbol (rising factorial) is defined by
( z ) n i = 1 n ( z + i 1 ) ,
with n N 0 . Note that, for all n N 0 , z N 0 , one has
( z ) n = Γ ( z + n ) Γ ( z ) , Γ ( z n ) = ( 1 ) n Γ ( z ) ( z + 1 ) n .
We will also need the binomial theorem ([3] (15.4.6))
2 F 1 a , b b ; z = ( 1 z ) a ,
Euler’s transformation (4), ([3] (15.8.1))
2 F 1 a , b c ; z = ( 1 z ) c a b 2 F 1 c a , c b c ; z ,
and the Gauss sum ([3] (15.4.20))
2 F 1 a , b c ; 1 = Γ ( c ) Γ ( c a b ) Γ ( c a ) Γ ( c b ) ,
for ( c a b ) > 0 . We will also use the generalized hypergeometric function
3 F 2 a , b , c d , e ; z n = 0 ( a ) n ( b ) n ( c ) n ( d ) n ( e ) n z n n ! ,
where d , e N 0 . We now produce a lemma for the Gauss hypergeometric function, which will be useful in our analysis of antiderivatives for associated Legendre functions of the second kind below.
Lemma 1.
Let z C \ [ 0 , ) . Then,
d d z 1 z ν + μ + 1 2 F 1 ν + μ + 1 2 , ν + μ + 2 2 ν + 3 2 ; 1 z 2 = ( ν + μ + 1 ) z ν + μ + 2 2 F 1 ν + μ + 2 2 , ν + μ + 3 2 ν + 3 2 ; 1 z 2 .
Proof. 
Differentiating the left-hand side of (6) using the chain rule and ([3] (15.5.1))
d d z 2 F 1 a , b c ; z = a b c 2 F 1 a + 1 , b + 1 c + 1 ; z ,
one produces an expression that involves the sum of two Gauss hypergeometric functions. One can then use the following Gauss relations for contiguous hypergeometric functions ([5] p. 58)
z 2 F 1 a + 1 , b + 1 c + 1 ; z = c a b 2 F 1 a , b + 1 c ; z 2 F 1 a + 1 , b c ; z ,
and [3] (15.5.12)
2 F 1 a , b + 1 c ; z = b a b 2 F 1 a , b c ; z + a b 2 F 1 a + 1 , b c ; z ,
to obtain the following formula
d d z 1 z a + b 1 2 2 F 1 a , b c ; 1 z 2 = a b + 1 2 z a + b + 1 2 2 F 1 a , b c ; 1 z 2 2 a z a + b + 1 2 2 F 1 a + 1 , b c ; 1 z 2 .
Because, for the Gauss hypergeometric function on the left-hand side of (6), a b + 1 2 = 0 , so the first term on the right-hand side of (10) vanishes and the lemma follows.  □
Definition 1.
Let z C , a , b C { , } . Define the following notations for nth iterated integrals of the functions f ( z ; a ) , g ( z ; b ) , respectively,
z b z b f ( w ; a ) ( d w ) n z b w n 1 b w 2 b w 1 b f ( w ; a ) d w d w 1 d w n 2 d w n 1 ,
a z a z g ( w ; b ) ( d w ) n : a z a w n 1 a w 2 a w 1 g ( w ; b ) d w d w 1 d w n 2 d w n 1 ,
where w 0 w , w n z , and a , b , are sets of fixed parameters.
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
d d z f μ ( z ) = λ μ f μ ± 1 ( z ) ,
where λ μ C * . Then, the following identity holds:
a x a x f μ ( w ) ( d w ) n = 1 λ μ 1 λ μ n k = n λ μ n λ μ n ± ( k 1 ) f μ n ± k ( a ) ( x a ) k k ! .
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
a x f μ ( w ) d w = 1 λ μ 1 a x d d w f μ 1 ( w ) d w = 1 λ μ 1 f μ 1 ( x ) f μ 1 ( a ) .
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:
a x a x f μ ( w ) ( d w ) n = 1 λ μ 1 a x a x f μ 1 ( w 1 ) f μ 1 ( a ) d w 1 d w n = 1 λ μ 1 λ μ ( 1 + n ) k = n + 1 λ μ ( n + 1 ) λ μ ( n + 1 ) ± ( k 1 ) f μ ( n + 1 ) ± k ( a ) ( x a ) k k ! ,
where we have used induction and the basic properties of integrals. Hence, the result follows.  □

3. 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] Section 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 ν μ : C \ ( , 1 ] C is defined as ([3] (14.3.6))
P ν μ ( z ) = 1 Γ ( 1 μ ) z + 1 z 1 μ / 2 2 F 1 ν , ν + 1 1 μ ; 1 z 2 .
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
P ν μ ( z ) = ( z 2 1 ) μ 2 2 μ Γ ( μ + 1 ) 2 F 1 ν + μ + 1 , ν + μ 1 + μ ; 1 z 2 .
The associated Legendre function of the second kind Q ν μ : C \ ( , 1 ] C can be defined in terms of the Gauss hypergeometric function as ([3] (14.3.10) and Section 14.21)
Q ν μ ( z ) π ( z 2 1 ) μ / 2 2 ν + 1 Γ ( ν + 3 2 ) z ν + μ + 1 2 F 1 ν + μ + 1 2 , ν + μ + 2 2 ν + 3 2 ; 1 z 2 ,
for | z | > 1 and, by analytic continuation of the Gauss hypergeometric function, elsewhere on z C \ ( , 1 ] .
Remark 1.
The normalized notation Q ν μ ( z ) is due to Olver ([8] p. 178) and it is defined in terms of the more commonly appearing Hobson notation for the associated Legendre function of the second kind Q ν μ ( z ) , as follows ([3] (14.3.10))
Q ν μ ( z ) = e i π μ Γ ( ν + μ + 1 ) Q ν μ ( z ) .
See ([3] Section 14.1) for more information on commonly appearing notations for the associated Legendre and Ferrers functions.
Remark 2.
Note that the following algebraic special cases for the associated Legendre function of the second kind, hold for μ = ν + 1 , ν + 2 , namely
Q ν ν + 1 ( z ) = z 2 1 z Q ν ν + 2 ( z ) = π 2 ν + 1 Γ ( ν + 3 2 ) ( z 2 1 ) ν + 1 2 ,
where we have used (16) and (3). Furthermore algebraic expressions for Q ν ν + n for all n N are obtainable from the order recurrence relation for associated Legendre functions of the second kind (cf. ([3] (14.10.6)))
Q ν ν + n + 2 ( z ) = 2 ( ν + n + 1 ) z ( 2 ν + n + 2 ) z 2 1 Q ν ν + n + 1 ( z ) n 2 ν + n + 2 Q ν ν + n ( z ) .
Because Q ν ν + 2 ( z ) is proportional to Q ν ν + 1 ( z ) , then all Q ν ν + m ( z ) is proportional to Q ν ν + 1 ( z ) for all m 2 .
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.
Theorem 1.
Let z C \ ( , 1 ] , μ C , ν = 3 2 n { 3 2 , 5 2 , } , n N 0 . Then,
Q 3 2 n μ ( z ) = ( 1 ) n π ( μ 2 1 4 ) ( 3 2 + μ ) n ( 3 2 μ ) n ( z 2 1 ) μ 2 2 n + 3 2 ( n + 1 ) ! z n + 3 2 + μ 2 F 1 3 4 + μ + n 2 , 5 4 + μ + n 2 n + 2 ; 1 z 2 .
Proof. 
Start with (16) then let ν = 3 2 n , followed by the application of ([3] First equation in Section 15.2(ii)),
lim c n 1 Γ ( c ) 2 F 1 a , b c ; z = ( a ) n + 1 ( b ) n + 1 ( n + 1 ) ! z n + 1 2 F 1 a + n + 1 , b + n + 1 n + 2 ; z ,
and the Pochhammer symbol identity for ν C , n N 0 ,
ν n 2 n ν + 1 2 n 2 n = ( 1 ) n 2 2 n ( 2 ν ) n ( 2 ν + 1 ) n ,
which follows from the duplication theorem for gamma functions [3] (5.5.5), and (2).  □
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.
Corollary 1.
Let z C \ ( , 1 ] , ν = 3 2 n { 3 2 , 5 2 , } , μ = ± ( 1 2 + k ) , n , k N 0 . Then
Q ν μ ( z ) = Q 3 2 n ± ( 1 2 + k ) ( z ) = 0 ,
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)
Q ν ± 1 2 ( cosh ξ ) = π 2 sinh ξ exp ( ( ν + 1 2 ) ξ ) Γ ( ν + 3 2 ) .
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 .
Lemma 3.
Let z C \ ( , 1 ] , μ C , ν = 3 2 n { 3 2 , 5 2 , } , n N 0 . Then,
Q ν μ ( z ) ( 1 ) ν 3 2 π 2 ν ( μ 2 1 4 ) Γ ( μ ν ) Γ ( μ ν ) z ν Γ ( 1 2 ν ) Γ ( 3 2 + μ ) Γ ( 3 2 μ ) .
Equivalently,
Q 3 2 n μ ( z ) ( 1 ) n π ( μ 2 1 4 ) ( 3 2 + μ ) n ( 3 2 μ ) n 2 n + 3 2 ( n + 1 ) ! z n + 3 2 .
Proof. 
The result follows by starting with (20) and examining its leading term behavior as z .  □

3.1. 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 z C \ ( , 1 ] , ν , μ C :
d d z Q ν μ ( z ) ( z 2 1 ) μ 2 = ( ν + μ + 1 ) ( z 2 1 ) μ + 1 2 Q ν μ + 1 ( z ) .
From this formula, the following antiderivative is obtained:
Q ν μ ( z ) ( z 2 1 ) μ 2 d z = Q ν μ 1 ( z ) ( ν + μ ) ( z 2 1 ) μ 1 2 + C ,
where C is an arbitrary constant.
Theorem 2.
Let z C \ ( , 1 ] , ν , μ C , such that ( ν + μ + 1 ) > 0 . Then,
Q ν μ ( z ) = ( ν + μ + 1 ) ( z 2 1 ) μ 2 z Q ν μ + 1 ( w ) ( w 2 1 ) μ + 1 2 d w .
Proof. 
Taking the limit of the antiderivative (25) evaluated at the endpoints of integration using the large argument asymptotics ([3] (14.8.15))
Q ν μ ( z ) π Γ ( ν + 3 2 ) ( 2 z ) ν + 1 , ν { 3 2 , 5 2 , 7 2 , } ,
and Lemma 3, which shows that Q ν μ ( z ) 0 as z for ν { 3 2 , 5 2 , 7 2 , } as well. Therefore, the integral is convergent, as indicated, which completes the proof.  □
Remark 5.
Iterating (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:
d n d z n Q ν μ ( z ) ( z 2 1 ) μ 2 = ( 1 ) n ( ν + μ + 1 ) n ( z 2 1 ) μ + n 2 Q ν μ + n ( z ) .
Theorem 3.
Let n N 0 , z C \ ( , 1 ] , n N 0 , ν , μ C , such that ( ν + μ n + 1 ) > 0 . Then,
z z Q ν μ ( w ) ( w 2 1 ) μ 2 ( d w ) n = ( 1 ) n Q ν μ n ( z ) ( ν μ ) n ( z 2 1 ) μ n 2 .
Proof. 
Iterating Theorem 2 with (1) while using induction with (1) completes the proof.  □
Remark 6.
It is clear that Theorem 3 is a generalization of cf. ([3] (14.6.8))
Q ν n ( z ) = ( 1 ) n ( ν ) n ( z 2 1 ) n 2 z z Q ν ( w ) ( d w ) n ,
by considering the μ = 0 specialization in (28), while using (2) and Hobson’s notation (see Remark 1).
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 z C \ ( , 1 ] , ν C \ { 1 2 , 3 2 , 5 2 , } ,
d z ( z 2 1 ) ν + 1 = 1 ( 2 ν + 1 ) z 2 ν + 1 2 F 1 ν + 1 2 , ν + 1 ν + 3 2 ; 1 z 2 + C = 2 ν Γ ( ν + 1 2 ) π ( z 2 1 ) ν 2 Q ν ν ( z ) + C ,
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.
Corollary 2.
Let ν C , such that ν > 1 2 , z C \ ( , 1 ] . Then,
Q ν ν ( z ) = Q ν ν ( z ) = π ( z 2 1 ) ν 2 2 ν Γ ( ν + 1 2 ) z d w ( w 2 1 ) ν + 1 .
Proof. 
Evaluating the antiderivative Theorem 7 at the endpoints of integration and taking advantage of ([3] (14.9.14))
Q ν μ ( z ) = Q ν μ ( z ) ,
completes the proof.  □

3.2. 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.
Corollary 3.
Let z C \ ( , 1 ] , ν , μ C . Then,
P ν 1 μ ( z ) = ( ν + μ ) ( z 2 1 ) ν 2 1 z P ν μ ( w ) ( w 2 1 ) ν + 2 2 d w .
Proof. 
Apply the Whipple formula ([3] (14.9.16))
Q ν μ ( z ) = π 2 ( z 2 1 ) 1 / 4 P μ 1 / 2 ν 1 / 2 z z 2 1 ,
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,
d n d z n P ν μ ( z ) ( z 2 1 ) μ 2 = ( 1 ) n ( ν + μ + 1 ) n ( μ ν ) n ( z 2 1 ) μ + n 2 P ν μ n ( z ) .
From the above result, integral representations can be obtained through repeated integration. For instance, the single integral result is given, as follows.
Corollary 4.
Let z C \ ( , 1 ] , ν , μ C . Then
1 z P ν μ ( w ) ( w 2 1 ) μ 2 d w = 1 ( ν + μ ) ( ν μ + 1 ) P ν μ + 1 ( z ) ( z 2 1 ) μ 1 2 1 2 μ 1 Γ ( μ ) .
Proof. 
In order to derive this result, after applying the fundamental theorem of calculus for some continuous function f on [ a , b ] ,
a b f ( t ) d t = f ( b ) f ( a ) ,
and taking advantage of ([3] (14.8.7))
lim z 1 + P ν μ ( z ) ( z 2 1 ) μ 2 = 1 2 μ Γ ( μ + 1 ) ,
this completes the proof.  □
The above result can be generalized by repeatedly integrating the above formula.
Theorem 4.
Let n N 0 , z C \ ( , 1 ] , ν , μ C . Then,
1 z 1 z P ν μ ( w ) ( w 2 1 ) μ 2 ( d w ) n = 1 ( ν μ ) n ( ν μ + 1 ) n ( 1 ) n P ν μ + n ( z ) ( z 2 1 ) μ n 2 ( μ ) n 2 μ n Γ ( μ + 1 ) k = 0 n 1 ( ν + μ + 1 n ) k ( μ ν n ) k k ! ( μ + 1 n ) k 1 z 2 k
= ( z 1 ) n 2 μ n ! Γ ( μ + 1 ) 3 F 2 ν + μ + 1 , μ ν , 1 μ + 1 , n + 1 ; 1 z 2 .
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))
P ν n ( z ) = ( z 2 1 ) n 2 1 z 1 z P ν ( w ) ( d w ) n ,
by considering the specialization μ = 0 in (37), which follows by using (2) and ([3] (14.9.13))
P ν n ( z ) = Γ ( ν n + 1 ) Γ ( ν + n + 1 ) P ν n ( z ) , n N 0 .
Remark 10.
Note the special value ([3] (14.5.19))
P ν ν ( z ) = ( z 2 1 ) ν 2 2 ν Γ ( ν + 1 ) .
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.
Corollary 5.
Let ν > 1 2 , z C \ ( , 1 ] . Then,
P ν ν ( z ) = 2 ν Γ ( ν + 1 2 ) π ( z 2 1 ) ν 2 + 2 ν + 1 π sin ( π ν ) Γ ( ν + 1 ) ( z 2 1 ) ν 2 z d w ( w 2 1 ) ν + 1 .
Proof. 
Start with the connection relation ([3] (14.9.15))
P ν μ ( z ) = Γ ( ν + μ + 1 ) Γ ( ν μ + 1 ) P ν μ ( z ) + 2 π sin ( π μ ) Γ ( ν + μ + 1 ) Q ν μ ( z ) ,
then, relying on (41), the choice μ = ν completes the proof.  □
Remark 11.
Observe that if ν = n N 0 , then
P n n ( z ) = 2 n Γ ( n + 1 2 ) ( z 2 1 ) n 2 π = ( 2 n 1 ) ! ! ( z 2 1 ) n 2 ,
where we have used ([4] (6.1.12)), and ( · ) ! ! is the double factorial symbol.
Corollary 6.
Let ν > 0 , z C \ ( , 1 ] . Then,
P ν ν ( z ) = 1 2 ν 1 Γ ( ν ) ( z 2 1 ) ν 2 1 z ( w 2 1 ) ν 1 d w .
Proof. 
Starting with (30) and using the Whipple relation for associated Legendre functions (32), followed by the application of the involution ([9] Section 2) ζ ( z ) log coth z 2 , then making the change of variables w = ζ / ζ 2 1 completes the proof.  □
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
P ν ν ( z ) = 1 Γ ( 2 ν + 1 ) P ν ν ( z ) 2 sin ( π ν ) π Q ν ν ( z ) .
After replacement of (30) and (42) in (41), we obtain
P ν ν ( z ) = sin ( π ν ) ( z 2 1 ) ν 2 π 2 ν 1 Γ ( ν + 1 2 ) z / z 2 1 d w ( w 2 1 ) ν + 1 / 2 + z d w ( w 2 1 ) ν + 1 = sin ( π ν ) ( z 2 1 ) ν 2 π 2 ν 1 Γ ( ν + 1 2 ) 1 d w ( w 2 1 ) ν + 1 = ( z 2 1 ) ν 2 2 ν Γ ( ν + 1 ) .
Which is simply a re-evaluation of (41). From the previous identities, the following result follows.
Corollary 7.
Let 1 2 ν 0 . Then,
Γ ( ν ) Γ ( ν + 1 2 ) 2 π = z / z 2 1 d w ( w 2 1 ) ν + 1 / 2 + z d w ( w 2 1 ) ν + 1 = 1 d w ( w 2 1 ) ν + 1 .
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.  □

4. 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 ν μ : ( 1 , 1 ) C are defined in ([3] (14.3.1))
P ν μ ( x ) 1 + x 1 x μ 2 1 Γ ( 1 μ ) 2 F 1 ν , ν + 1 1 μ ; 1 x 2 ,
= ( 1 x 2 ) μ 2 2 μ Γ ( 1 + μ ) 2 F 1 ν + μ + 1 , μ ν 1 + μ ; 1 x 2 ,
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. Additionally, Q ν μ : ( 1 , 1 ) C is defined in ([3] (14.3.2))
Q ν μ ( x ) π 2 sin ( π μ ) cos ( π μ ) Γ ( 1 μ ) 1 + x 1 x μ 2 2 F 1 ν , ν + 1 1 μ ; 1 x 2 Γ ( ν + μ + 1 ) Γ ( ν μ + 1 ) 1 x 1 + x μ 2 1 Γ ( 1 + μ ) 2 F 1 ν , ν + 1 1 + μ ; 1 x 2 ,
where μ Z . However, Q ν μ ( x ) can be continued analytically for μ Z , which is demonstrated by ([3] (14.3.12)),
Q ν μ ( x ) = π 2 μ 1 ( 1 x 2 ) μ 2 sin π 2 ( ν + μ ) Γ ( ν + μ + 1 2 ) Γ ( ν μ + 2 2 ) 2 F 1 ν + μ 2 , ν μ + 1 2 1 2 ; x 2 + 2 cos π 2 ( ν + μ ) Γ ( ν + μ + 2 2 ) x Γ ( ν μ + 1 2 ) 2 F 1 ν μ + 1 2 , ν μ + 2 2 3 2 ; x 2 .
Another hypergeometric representation of the Ferrers function of the second kind that we will use below is
Q ν μ ( x ) = 2 μ 1 cos ( π μ ) ( 1 x 2 ) μ 2 Γ ( μ ) x ν + μ 2 F 1 ν μ 2 , ν μ + 1 2 1 μ ; x 2 1 x 2 + Γ ( ν + μ + 1 ) Γ ( μ ) 2 μ + 1 Γ ( ν μ + 1 ) ( 1 x 2 ) μ 2 x ν μ 2 F 1 μ ν 2 , μ ν + 1 2 μ + 1 ; x 2 1 x 2 ,
which can be obtained by a limiting procedure ([3] cf. (14.23.5))
Q ν μ ( x ) = Γ ( ν + μ + 1 ) 2 e 1 2 i π μ Q ν μ ( x + i 0 ) + e 1 2 i π μ Q ν μ ( x i 0 ) ,
for x ( 1 , 1 ) , starting from ([10] Entry 29, p. 162).

4.1. 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 .
Remark 12.
Using (7), (46) produces the following derivative formula for n N 0 , x ( 1 , 1 ) , ν , μ C , namely
d n d x n P ν μ ( x ) ( 1 x 2 ) μ 2 = ( 1 ) n ( ν + μ + 1 ) n ( μ ν ) n P ν μ n ( x ) ( 1 x 2 ) μ + n 2 .
From (46), integral representations can be obtained through repeated integration. For instance, the single integral result is given, as follows.
Corollary 8.
Let x ( 1 , 1 ) , ν , μ C . Then,
x 1 P ν μ ( w ) ( 1 w 2 ) μ 2 d w = 1 ( ν + μ ) ( ν μ + 1 ) 1 2 μ 1 Γ ( μ ) P ν μ + 1 ( x ) ( 1 x 2 ) μ 1 2 .
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))
P ν μ ( x ) 1 Γ ( 1 μ ) 2 1 x μ 2 ,
as x 1 , which completes the proof.  □
The above result can be generalized by repeatedly integrating the above formula.
Theorem 5.
Let n N 0 , x ( 1 , 1 ) , ν , μ C . Then,
x 1 x 1 P ν μ ( w ) ( 1 w 2 ) μ 2 ( d w ) n = 1 ( ν μ ) n ( ν μ + 1 ) n P ν μ + n ( x ) ( 1 x 2 ) μ n 2 ( 1 ) n ( μ ) n 2 μ n Γ ( μ + 1 ) k = 0 n 1 ( ν + μ + 1 n ) k ( μ ν n ) k k ! ( μ + 1 n ) k 1 x 2 k
= ( 1 x ) n 2 μ n ! Γ ( μ + 1 ) 3 F 2 ν + μ + 1 , μ ν , 1 μ + 1 , n + 1 ; 1 x 2 .
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))
P ν n ( x ) = ( 1 x 2 ) n 2 x 1 x 1 P ν ( w ) ( d w ) n ,
by considering the specialization μ = 0 in (53), where we have used ([3] (14.9.3))
P ν n ( x ) = ( 1 ) n Γ ( ν n + 1 ) Γ ( ν + n + 1 ) P ν n ( x ) , n N 0 .
By using the antiderivative ([3] (14.17.2))
P ν μ ( w ) ( 1 w 2 ) μ 2 d w = 1 ( ν + μ ) ( ν μ + 1 ) P ν μ + 1 ( x ) ( 1 x 2 ) μ 1 2 + C ,
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,
0 x P ν μ ( w ) ( 1 w 2 ) μ 2 d w = 1 ( ν + μ ) ( ν μ + 1 ) P ν μ + 1 ( x ) ( 1 x 2 ) μ 1 2 π 2 μ 1 Γ ν + μ + 1 2 Γ μ ν 2 .
Proof. 
Evaluating (57) at the endpoints of integration while using ([3] (14.5.1))
P ν μ ( 0 ) = π 2 μ Γ ν + μ + 2 2 Γ μ ν + 1 2 ,
completes the proof.  □
Theorem 7.
Let n N 0 , x ( 1 , 1 ) , ν , μ C . Then,
0 x 0 x P ν μ ( w ) ( 1 w 2 ) μ 2 ( d w ) n = ( 1 ) n k = n ( ν + μ + 1 ) k n ( μ ν ) k n P ν μ + n k ( 0 ) = ( 1 ) n ( ν μ ) n ( ν μ + 1 ) n
× P ν μ + n ( x ) ( 1 x 2 ) μ n 2 π 2 μ n k = 0 n 1 ( ν + μ n + 1 ) k ( μ ν n ) k x 2 k k ! Γ ν + μ + 2 n + k 2 Γ μ ν + 1 n + k 2
= π x n 2 μ n ! Γ ν + μ + 2 2 Γ μ ν + 1 2 3 F 2 μ ν 2 , ν + μ + 1 2 , 1 n + 1 2 , n + 2 2 ; x 2 π x n + 1 2 μ 1 ( n + 1 ) ! Γ ν + μ + 1 2 Γ μ ν 2 3 F 2 μ ν + 1 2 , ν + μ + 2 2 , 1 n + 2 2 , n + 3 2 ; x 2 .
Proof. 
Repeatedly applying Theorem 6 without evaluating P ν μ ( 0 ) and then computing the Maclaurin expansion of P ν μ + n ( x ) / ( 1 x 2 ) ( μ n ) / 2 yields the first expression. Using induction evaluating P ν μ + n ( 0 ) with (1) produces the second expression. The third expression is obtained by starting with the first expression, evaluating P ν μ + n k ( 0 ) , shifting the sum index by n, and splitting the sum into even and odd parts. On the other hand, by applying the antiderivative
On the other hand, by applying the antiderivative ([3] (14.17.1))
P ν μ ( w ) ( 1 w 2 ) μ 2 d w = P ν μ 1 ( x ) ( 1 x 2 ) μ 1 2 + C ,
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 ,
d n d x n P ν μ ( x ) ( 1 x 2 ) μ 2 = ( 1 ) n P ν μ + n ( x ) ( 1 x 2 ) μ + n 2 .
An evaluation of (63) at the endpoints of integration produces the following result.
Theorem 8.
Let x ( 1 , 1 ) , ν , μ C , μ > 0 . Then,
x 1 ( 1 w 2 ) μ 2 P ν μ ( w ) d w = ( 1 x 2 ) μ + 1 2 P ν μ 1 ( x ) .
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)
( 1 x 2 ) μ 2 P ν μ ( x ) 0 ,
as x 1 . This completes the proof.  □
Theorem 9.
Let x ( 1 , 1 ) , ν , μ C , μ > 0 . Then,
t o x 1 x 1 ( 1 w 2 ) μ 2 P ν μ ( w ) ( d w ) n = ( 1 x 2 ) μ + n 2 P ν μ n ( x ) .
Proof. 
Repeatedly applying Theorem 8 through induction proves the result.  □
Theorem 10.
Let n N 0 , x ( 1 , 1 ) , ν , μ C . Then,
0 x P ν μ ( w ) ( 1 w 2 ) μ 2 d w = P ν μ 1 ( x ) ( 1 x 2 ) μ 1 2 + 2 μ 1 π Γ ν μ + 3 2 Γ ν μ + 2 2 .
Proof. 
Evaluating (57) at the endpoints of integration while using (59) completes the proof.  □
Theorem 11.
Let x ( 1 , 1 ) , ν , μ C . Then,
0 x 0 x P ν μ ( w ) ( 1 w 2 ) μ 2 ( d w ) n = ( 1 ) n k = n ( x ) k k ! P ν μ n + k ( 0 ) = ( 1 ) n P ν μ n ( x ) ( 1 x 2 ) μ n 2 2 μ n π k = 0 n 1 ( 2 x ) k k ! Γ ν μ + 2 + n k 2 Γ ν μ + 1 + n k 2 = π 2 μ x n n ! Γ ν μ + 2 2 Γ ν μ + 1 2 3 F 2 μ ν 2 , ν + μ + 1 2 , 1 n + 1 2 , n + 2 2 ; x 2 π 2 μ + 1 x n + 1 ( n + 1 ) ! Γ ν μ + 1 2 Γ ν μ 2 3 F 2 μ ν + 1 2 , ν + μ + 2 2 , 1 n + 2 2 , n + 3 2 ; x 2 .
Proof. 
Repeatedly applying Theorem 10 without evaluating P ν μ ( 0 ) and then computing the Maclaurin expansion of P ν μ n ( x ) / ( 1 x 2 ) ( μ n ) / 2 yields the first expression. Using induction evaluating P ν μ n ( 0 ) with (1) produces the second expression. The third expression is obtained by starting with the first expression, evaluating P ν μ n + k ( 0 ) , shifting the sum index by n, and splitting the sum into even and odd parts.  □
A definite-integral result near the singularity at x = 1 follows using (97), (98) and (5), namely
0 1 ( 1 w 2 ) ν 1 d w = π Γ ( ν ) 2 Γ ( ν + 1 2 ) ,
for ν > 0 . The well-known special value (see ([3] (14.5.18)))
P ν ν ( x ) = ( 1 x 2 ) ν 2 2 ν Γ ( ν + 1 ) ,
in conjunction with ([11] (8.737.1)), yields the following integral representation.
Corollary 9.
Let ν C , x ( 1 , 1 ) . Then,
P ν ν ( x ) = 2 ν ( 1 x 2 ) ν 2 π Γ ( ν + 1 2 ) cos ( π ν ) + 2 Γ ( ν + 1 ) π sin ( π ν ) 0 x d w ( 1 w 2 ) ν + 1 .
Proof. 
Start with ([3] (14.9.2))
P ν μ ( x ) = cos ( π μ ) Γ ( ν + μ + 1 ) Γ ( ν μ + 1 ) P ν μ ( x ) + 2 π sin ( π μ ) Γ ( ν + μ + 1 ) Γ ( ν μ + 1 ) Q ν μ ( x ) ,
replace μ = ν , then using (98), (69) completes the proof.  □
Remark 15.
Note that, if ν = n N 0 , then
P n n ( x ) = ( 2 ) n Γ ( n + 1 2 ) ( 1 x 2 ) n 2 π = ( 1 ) n ( 2 n 1 ) ! ! ( 1 x 2 ) n 2 ,
where we have used ([4] (6.1.12)).

4.2. 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 ν μ : ( 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 .
Lemma 4.
Let x ( 1 , 1 ) , ν , μ C , such that μ N , ν μ N 0 . Then,
x 1 ( 1 w 2 ) μ 2 Q ν μ ( w ) d w = ( 1 x 2 ) μ + 1 2 Q ν μ 1 ( x ) 2 μ Γ ( μ + 1 ) Γ ( ν μ ) Γ ( ν + μ + 2 ) .
Proof. 
The Ferrers function of the second kind as x approaches the singularity at x = 1 has the following behavior ([3] (14.8.6))
( 1 x 2 ) μ 2 Q ν μ ( x ) 2 μ 1 Γ ( μ ) Γ ( ν μ + 1 ) Γ ( ν + μ + 1 ) ,
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
d n d x n ( 1 x 2 ) μ 2 Q ν μ ( x ) = ( 1 ) n ( 1 x 2 ) μ n 2 Q ν μ + n ( x ) .
Theorem 12.
Let n N 0 , x ( 1 , 1 ) , ν , μ C . Then,
x 1 x 1 ( 1 w 2 ) μ 2 Q ν μ ( w ) ( d w ) n = ( 1 x 2 ) μ + n 2 Q ν μ n ( x ) ( 1 ) n 2 μ + n 1 Γ ( μ ) Γ ( ν μ + 1 ) ( μ ) n Γ ( ν + μ + 1 ) ( μ ν ) n ( ν + μ + 1 ) n k = 0 n 1 ( ν μ n + 1 ) k ( ν μ n ) k k ! ( μ n + 1 ) k 1 x 2 k
= π 2 cot ( π μ ) ( 1 x 2 ) μ + n 2 P ν μ n ( x ) + 2 μ 1 Γ ( μ ) Γ ( ν μ + 1 ) ( 1 x ) n n ! Γ ( ν + μ + 1 ) 3 F 2 ν μ + 1 , ν μ , 1 n + 1 , 1 μ ; 1 x 2 .
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
x 1 x 1 Q ν ( w ) ( d w ) n = ( 1 x 2 ) n 2 Q ν n ( x ) ( 1 ) n 2 n 1 ( n 1 ) ! ( ν ) n ( ν + 1 ) n k = 0 n 1 ( ν n + 1 ) k ( ν n ) k k ! ( 1 n ) k 1 x 2 k .
Now, we present a similar result for the Ferrers function of the second kind with order μ instead of μ .
Lemma 5.
Let x ( 1 , 1 ) , ν , μ C , such that μ N , ν μ N 0 . Then,
x 1 ( 1 w 2 ) μ 2 Q ν μ ( w ) d w = 1 ( μ ν ) ( ν + μ + 1 ) ( 1 x 2 ) μ + 1 2 Q ν μ + 1 ( x ) + 2 μ π Γ ( μ + 1 ) Γ ( μ 3 2 ) Γ ( μ + 5 2 ) .
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)))
( 1 x 2 ) μ 2 Q ν μ ( x ) 2 μ 1 π Γ ( μ ) Γ ( μ 1 2 ) Γ ( μ + 3 2 ) ,
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
d n d x n ( 1 x 2 ) μ 2 Q ν μ ( x ) = ( 1 ) n ( ν μ + 1 ) n ( ν μ ) n ( 1 x 2 ) μ n 2 Q ν μ n ( x ) .
Theorem 13.
Let n N 0 , x ( 1 , 1 ) , ν , μ C . Then,
x 1 x 1 ( 1 w 2 ) μ 2 Q ν μ ( w ) ( d w ) n = 1 ( μ ν ) n ( ν + μ + 1 ) n ( 1 x 2 ) μ + n 2 Q ν μ + n ( x ) + ( 1 ) n 1 2 μ + n 1 cos ( π μ ) Γ ( μ + n ) k = 0 n 1 ( ν μ n + 1 ) k ( ν μ n ) k k ! ( μ n + 1 ) k 1 x 2 k
= π Γ ( ν + μ + 1 ) 2 Γ ( ν μ + 1 ) sin ( π μ ) ( 1 x 2 ) μ + n 2 P ν μ n ( x ) + 2 μ 1 cos ( π μ ) Γ ( μ ) n ! ( 1 x ) n 3 F 2 ν μ + 1 , ν μ , 1 n + 1 , 1 μ ; 1 x 2 .
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.
Remark 20.
Taking μ 0 limit in Theorem 13 produces the following multi-integration result for x ( 1 , 1 ) , ν , μ C , namely
x 1 x 1 Q ν ( w ) ( d w ) n = 1 ( ν ) n ( ν + 1 ) n ( 1 x 2 ) n 2 Q ν n ( x ) + ( 1 ) n 1 2 n 1 ( n 1 ) ! k = 0 n 1 ( ν n + 1 ) k ( ν n ) k k ! ( 1 n ) k 1 x 2 k .
Remark 21.
Note that, in Theorems 12 and 13, it is tempting to consider the μ 0 limit while using their 3 F 2 representations. However, to zeroth order in μ, the limits cancel. One must then determine a first order approximation in μ in order to determine the limit behavior. After performing this calculation in both of these situations, then it turns out that the result is given in terms of the sum of several double hypergeometric series of Kampé de Férier type (see e.g., ([12] p. 27)). Because this result is very cumbersome and does not really shed much light on these limits, we have instead presented the above Remarks 17 and 20. It should also be pointed out regarding the fact that there does not seem to be an analogous formula for the Ferrers function of the second kind in the classical list ([3] (14.6.6–8)), the above reasoning most likely explains this fact. The conclusion is that any formula for negative integer order Ferrers functions of the second kind will involve a finite sum of polynomial terms in addition to the multi-integral of Q ν ( x ) , as indicated in Remarks 17 and 20.
Next, we treat some multi-integrals of the Ferrers function of the second kind from the origin.
Remark 22.
Applying the fundamental theorem of calculus (35) to ([3] (14.17.2)) produces the following derivative formula for n N 0 , x ( 1 , 1 ) , ν , μ C , namely
d n d x n Q ν μ ( x ) ( 1 x 2 ) μ 2 = ( 1 ) n ( ν + μ + 1 ) n ( μ ν ) n Q ν μ n ( x ) ( 1 x 2 ) μ + n 2 .
Theorem 14.
Let x ( 1 , 1 ) , ν , μ C . Then,
0 x Q ν μ ( w ) ( 1 w 2 ) μ 2 d w = 1 ( ν + μ ) ( ν μ + 1 ) Q ν μ + 1 ( x ) ( 1 x 2 ) μ 1 2 π 3 2 Γ ν μ + 2 2 2 μ Γ ν + μ + 1 2 Γ μ ν 1 2 Γ ν μ + 3 2 .
Proof. 
Evaluating ([3] (14.17.2)) (expressed as a Ferrers function of the second kind) at the endpoints of integration using ([3] (14.5.3))
Q ν μ ( 0 ) = π 3 2 Γ ν μ + 1 2 2 μ + 1 Γ ν + μ + 2 2 Γ μ ν 2 Γ ν μ + 2 2 ,
completes the proof.  □
Theorem 15.
Let n N 0 , x ( 1 , 1 ) , ν , μ C . Then,
0 x 0 x Q ν μ ( w ) ( 1 w 2 ) μ 2 ( d w ) n = ( 1 ) n k = n ( x ) k k ! ( ν + μ + 1 ) k n ( μ ν ) k n Q ν μ + n k ( 0 )
= ( 1 ) n ( ν μ ) n ( ν μ + 1 ) n Q ν μ + n ( x ) ( 1 x 2 ) μ n 2 π 3 2 2 μ n + 1 k = 0 n 1 ( ν + μ + 1 n ) k ( μ ν n ) k Γ ν μ + 1 + n k 2 x 2 k k ! Γ ν + μ + 2 n + k 2 Γ μ ν n + k 2 Γ ν μ + 2 + n k 2
= π 3 2 x n Γ ν μ + 1 2 n ! 2 μ + 1 Γ ν + μ + 2 2 Γ μ ν 2 Γ ν μ + 2 2 3 F 2 ν + μ + 1 2 , μ ν 2 , 1 n + 1 2 , n + 2 2 ; x 2 + π 3 2 x n + 1 Γ ν μ + 2 2 ( n + 1 ) ! 2 μ Γ ν + μ + 1 2 Γ μ ν + 1 2 Γ ν μ + 1 2 3 F 2 ν + μ + 2 2 , μ ν + 1 2 , 1 n + 2 2 , n + 3 2 ; x 2 .
Proof. 
Repeatedly applying Theorem 14 without evaluating Q ν μ ( 0 ) and then computing the Maclaurin expansion of Q ν μ + n ( x ) / ( 1 x 2 ) ( μ n ) / 2 yields the first expression. Using induction evaluating Q ν μ + n ( 0 ) with (1) produces the second expression. The third expression is obtained by starting with the first expression, evaluating Q ν μ + n k ( 0 ) , shifting the sum index by n and splitting the sum into even and odd parts.  □
Remark 23.
Taking the limit as μ = 0 in Theorem 15 produces the following multi-integration result, namely for x ( 1 , 1 ) , ν , μ C , then
0 x 0 x Q ν ( w ) ( d w ) n = 1 ( ν ) n ( ν + 1 ) n ( 1 ) n ( 1 x 2 ) n 2 Q ν n ( x ) + ( 1 ) n + 1 2 n 1 π 3 2 k = 0 n 1 ( ν + 1 n ) k ( ν n ) k Γ ν + 1 + n k 2 x 2 k k ! Γ ν + 2 n + k 2 Γ ν n + k 2 Γ ν + 2 + n k 2 .
On the other hand, by applying the antiderivative ([3] (14.17.1))
Q ν μ ( w ) ( 1 w 2 ) μ 2 d w = Q ν μ 1 ( x ) ( 1 x 2 ) μ 1 2 + C ,
where C is an arbitrary constant for deriving some interesting integral representations for Ferrers functions of the second kind. An examination of the above formula (91) produces the following results. For instance, by applying this formula to obtain a useful derivative formula for Ferrers functions of the second kind.
Remark 24.
Differentiating the above formula with (1) produces the following formula for x ( 1 , 1 ) , ν , μ C ,
d n d x n Q ν μ ( x ) ( 1 x 2 ) μ 2 = ( 1 ) n Q ν μ + n ( x ) ( 1 x 2 ) μ + n 2 .
Theorem 16.
Let x ( 1 , 1 ) , ν , μ C , such that ν + μ 2 N 0 . Then,
0 x Q ν μ ( w ) ( 1 w 2 ) μ 2 d w = Q ν μ 1 ( x ) ( 1 x 2 ) μ 1 2 2 μ 2 π 3 2 Γ ν + μ 2 Γ ν μ + 3 2 Γ ν μ 1 2 Γ ν + μ + 3 2 .
Proof. 
Evaluating (91) at the endpoints of integration while using (86) completes the proof.  □
Theorem 17.
Let n N 0 , x ( 1 , 1 ) , ν , μ C . Then,
0 x 0 x Q ν μ ( w ) ( 1 w 2 ) μ 2 ( d w ) n = ( 1 ) n k = n ( x ) k k ! Q ν μ n + k ( 0 ) = ( 1 ) n Q ν μ n ( x ) ( 1 x 2 ) μ n 2 + ( 1 ) n π 3 2 2 n + 1 μ k = 0 n 1 Γ ν + μ + 1 n + k 2 ( 2 x ) k k ! Γ ν μ + 2 + n k 2 Γ ν μ 2 + n k 2 Γ ν + μ + 4 n + k 2 = π 3 2 2 μ 1 x n Γ ν + μ + 1 2 n ! Γ ν μ + 2 2 Γ ν μ 2 Γ ν + μ + 2 2 3 F 2 μ ν 2 , ν + μ + 1 2 , 1 n + 1 2 , n + 2 2 ; x 2
π 3 2 2 μ x n + 1 Γ ν + μ + 2 2 ( n + 1 ) ! Γ ν μ + 1 2 Γ ν μ 1 2 Γ ν + μ + 3 2 3 F 2 μ ν + 1 2 , ν + μ + 2 2 , 1 n + 2 2 , n + 3 2 ; x 2 .
Proof. 
Repeatedly applying Theorem 16 without evaluating Q ν μ ( 0 ) and then computing the Maclaurin expansion of Q ν μ n ( x ) / ( 1 x 2 ) ( μ n ) / 2 yields the first expression. Using induction evaluating Q ν μ n ( 0 ) with (1) produces the second expression. The third expression is obtained by starting with the first expression, evaluating Q ν μ n + k ( 0 ) , shifting the sum index by n, and splitting the sum into even and odd parts.  □
Theorem 18.
Let ν C . Then,
d x ( 1 x 2 ) ν + 1 = x 2 F 1 1 2 , ν + 1 3 2 ; x 2 + C = 2 ν Γ ( ν + 1 2 ) π ( 1 x 2 ) ν 2 Q ν ν ( x ) + C ,
where C is an arbitrary constant.
Proof. 
The Gauss hypergeometric function in the antiderivative follows using (7)–(9), as in the proof of Theorem 7, with the Ferrers function directly following using
Q ν ν ( x ) = π x ( 1 x 2 ) ν 2 2 ν Γ ( ν + 1 2 ) 2 F 1 1 2 , ν + 1 3 2 ; x 2 ,
which follows from (48). This completes the proof.  □
The following very simple integral representation for the Ferrers function of the second kind is a consequence of Theorem 18.
Corollary 10.
Let ν C , x ( 1 , 1 ) . Then,
Q ν ν ( x ) = π ( 1 x 2 ) ν 2 2 ν Γ ( ν + 1 2 ) 0 x d w ( 1 w 2 ) ν + 1 .
Proof. 
Evaluating the antiderivative Theorem 18 at the endpoints of integration completes the proof.  □
Remark 25.
One can also show that Corollary 10 also directly follows from Theorem 14. This is true, even though Theorem 14 is not strictly valid for μ = ν . The result can be obtained by taking the limit as μ ν in Theorem 14 and using the Gauss hypergeometric representation of the Ferrers function of the second kind with argument ( 1 x 2 ) , namely (49).
Corollary 11.
Let ν C , x ( 1 , 1 ) . Then,
Q ν ν ( x ) = 2 ν 1 π Γ ( ν + 1 2 ) sin ( π ν ) ( 1 x 2 ) ν 2 + 2 ν Γ ( ν + 1 ) cos ( π ν ) ( 1 x 2 ) ν 2 0 x d w ( 1 w 2 ) ν + 1 .
Proof. 
Using the connection relation ([10] p. 170)
Q ν μ ( x ) = Γ ( ν μ + 1 ) Γ ( ν + μ + 1 ) cos ( π μ ) Q ν μ ( x ) + π 2 sin ( π μ ) P ν μ ( x ) ,
setting μ = ν , and using the above results completes the proof.  □
Remark 26.
Note that, if ν = n + 1 2 , n N 0 , then Corollary 11 reduces to the following special value
Q n + 1 2 n + 1 2 ( x ) = ( 1 ) n + 1 2 n 1 2 n ! π ( 1 x 2 ) n 2 + 1 4 .

5. Conclusions

In this paper, we explore some implications of the existence of multi-derivative formulae for associated Legendre functions of the first and second kinds P ν μ , Q ν μ , and Ferrers functions of the first and second kinds P ν μ , Q ν μ . These multi-derivative formulae (see Remarks 5, 8, 12, 14, 16, 18, 22 and 24) have the useful property that the degree ( ν ) is left unchanged by the multi-derivative. The order ( μ ) is then shifted by unit increments, depending on the number of derivatives. These multi-derivative formulae generalize some classical multi-derivative formulae for these functions with integer order ([3] (14.6.1)–(14.6.5)).
Because of the existence of these multi-derivative formulae, and certain special known values and limiting behaviors near the singularities of these functions, we derive several multi-integral representations for these functions. These multi-integral representations are shown to be given either in terms of (i) a sum of two 3 F 2 ’s (Theorems 7, 9, 15 and 17); (ii) a 2 F 1 and a 3 F 2 (Theorems 12 and 13); (iii) a single 3 F 2 (Theorems 4 and 5); or (iv) a single 2 F 1 (Theorems 3 and 9). These multi-integral representations generalize some classical multi-integrals for these functions with integer order ([3] (14.6.6)–(14.6.8)].
Many of the functions encountered in this work represent fundamental solutions for the Laplace–Beltrami operator on Riemannian manifolds of constant curvature, as mentioned in the introduction. Multi-integrals and derivatives of these functions are essential in performing global analysis for these fundamental solutions on these manifolds. One interesting open problem where this work is almost certainly essential is for obtaining fundamental solutions of natural powers of the Laplace–Beltrami operator (polyharmonic) on these manifolds. This analysis will be investigated in future publications.

Author Contributions

H.S.C. and R.S.C.-S. conceived the mathematics; H.S.C. and R.S.C.-S. wrote the paper. All authors have read and agreed to the published version of the manuscript.

Funding

The research of R.S.C.-S. was funded by Agencia Estatal de Investigación of Spain, grant number PGC-2018-096504-B-C33.

Conflicts of Interest

The authors declare no conflict of interest.

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