## Abstract

Associated Legendre functions of fractional degree appear in the solution of boundary value problems on wedges or in toroidal geometries, and elsewhere in applied mathematics. In the classical case when the degree is half an odd integer, they can be expressed using complete elliptic integrals. In this study, many transformations are derived, which reduce the case when the degree differs from an integer by one-third, one-fourth or one-sixth to the classical case. These transformations or identities facilitate the symbolic manipulation and evaluation of Legendre and Ferrers functions. They generalize both Ramanujan's transformations of elliptic integrals and Whipple's formula, which relates Legendre functions of the first and second kinds. The proofs employ algebraic coordinate transformations, specified by algebraic curves.

## 1. Introduction

In applied mathematics and theoretical physics, Legendre or associated Legendre functions occur widely. Their properties are summarized in many places [1,2]. The most familiar are *z*∈(−1,1), or more generally on the complex *z*-plane with cuts *z*-plane with cut *ν* and order *μ*, satisfy the same second-order ordinary differential equation with parameters *ν* and *μ*, and in the absence of cuts may be multivalued.

The Ferrers functions *n* and integral order *m* are the most familiar. Spherical harmonics _{n}(*z*), is the *n*th Legendre polynomial. It needs no cuts for single-valuedness, and, in fact, *P*_{n} equals P_{n}. If, on the other hand, *m* is odd, then *θ*≤*π* and 0≤*ϕ*<2*π*, is what causes the quantization of the degree and order to integers. It should be noted that Ferrers functions of half-odd-integer degree and order find application in quantum mechanics [3]. Like the Legendre polynomials, they are elementary functions of *z*.

Ferrers or Legendre functions in which the degree *ν* is a half-odd-integer and the order is an integer are not elementary but can be expressed in terms of the complete elliptic integrals *K*=*K*(*m*), *E*=*E*(*m*), where *m* (often denoted *k*^{2}) is the elliptic modular parameter. (For instance, P_{−1/2}(*z*) equals (2/*π*)*K*((1−*z*)/2).) Ferrers or Legendre functions of *unrestricted* degree *ϕ*:
*ν* of a generating function for spherical harmonics [4], ch. VII, §7.3. Equation (1.1b) is a ‘generalized Heine identity’ which has attracted attention [5], and can be viewed as an analytic continuation of (1.1a). When **x**−**x**′| potential [6,7]. Equation (1.1b) also appears in celestial mechanics, in the analysis (originally) of planetary perturbations [8]. The coefficient of the Fourier mode *s*>0 is a half-odd integer, is called a Laplace coefficient. By (1.1b), it can be expressed in terms of the Legendre functions

Legendre (rather than Ferrers) functions with *ν* a half-odd integer and *μ* an integer are commonly called toroidal or ‘anchor ring’ functions, because harmonics, including factors of the form

### (a) Overview of results

It is shown that Legendre and Ferrers functions of any degree *ν* differing from an integer by ±1/*r*, for *r*=3,4,6, can be expressed in terms of like functions of half-odd integer degree. (The order here must be an integer.) This statement, which leads to unexpected closed-form expressions in terms of complete elliptic integrals, is one consequence of the main results, the Legendre identities in §3, which facilitate the rewriting and evaluation of *ν* differs from −1/*r* or +1/*r* by a non-zero integer is handled by applying differential recurrences, to shift the degree.) The most striking identities may be
*x*∈(0,1). This is the little known Fourier expansion of *ν* differs from an integer by ±1/*r* with *r*=3,4,6 (as well as the classical case *r*=2), the Fourier coefficients of

Identities such as (1.2a),(1.2b) and the full collection in §3 are closely related to certain function transformations of Ramanujan. In his famous notebooks [15], ch. 33, he developed a theory of elliptic integrals with non-classical ‘signature’ *r*=3,4,6 and related them to the classical integrals, which have signature 2. His theory yields formulae for the Ferrers functions P_{−1/r} in terms of P_{−1/2}, or equivalently the classical complete elliptic integral *K*, with an algebraically transformed argument. (For a compact list of Ramanujan's transformation formulae that can be written in this way, see [16], lemma 2.1.) The identities derived here include several of Ramanujan's rationally parametrized formulae, but such identities as (1.2a),(1.2b) are more general, in that they are formulae for Ferrers or Legendre functions of arbitrary (i.e. non-zero) order −*α*. The parametrization by trigonometric functions is another novel feature.

### (b) Methods

The technique used below for deriving Legendre identities was developed by considering Whipple's well-known *L*=(*p*^{2}+1)/(*p*^{2}−1) and *R*= (*p*^{2}+1)/2*p*, being algebraically related by (*L*^{2}−1)(*R*^{2}−1)=1. This relation defines an algebraic curve, which is parametrized by *p*, though it could be parametrized as *L*=*z* and

The point is that the correspondence *p*, and as well by the left-hand side with *same* element of the (two-dimensional) solution space of *ν*=−1/*r*, *r*=3,4,6, to those of other degrees, are all derived in a similar way, from algebraic curves.

### (c) Applications

Legendre functions of fractional degree occur in many areas of applied mathematics. One lies in mathematical physics: the representation theory of certain Lie algebras [18]. Another is geometric–analytic: the spectral analysis of Laplacian-like operators on spaces of negative curvature, which is of interest because of its connection to quantum chaos [19]. If Δ_{LB} denotes the Laplace–Beltrami operator on the real hyperbolic space _{LB}+*κ*^{2})^{−1}(**x**,**x**′) will be proportional to *d* is the hyperbolic distance between **x**,**x**′ and the degree *ν* depends on *κ*^{2}. If *κ*^{2}=0, then *ν*=*n*/2−1; and more generally, *ν* equals *κ*^{2}=0 case.) It follows that this Green's function on *κ*^{2}<0.

Another notable application area is the Tricomi problem, which occurs in two-dimensional transonic potential flow [23,24]. The Tricomi differential equation on the *θ*–*η* (i.e. hodograph) plane, *P*_{−1/6}. (See [26,27] and [28], ch. 13.) In fact, the so-called Gellerstedt generalization *P*_{−1/r},*Q*_{−1/r}, where *r*=3,4,6 for *k*=4,2,1 [29]. Applying such identities as (1.2a),(1.2b) will express such solutions in terms of complete elliptic integrals.

An additional application area is classical: the separation of variables in boundary value problems, posed on wedge-shaped domains. Along this line, Fock [30] used toroidal coordinates in solving a problem on a wedge of opening angle 3*π*/2, and was led to the fractional-degree functions *P*_{−1/6},*Q*_{−1/6}. More recently, a magnetostatic potential has been expanded near a cubic corner in modes of the form *ν* is corner-specific and known only numerically [31]. If one opening angle of the corner is increased to *π*, then it becomes a right-angled wedge, and the appropriate *ν* becomes fractional. In such situations, closed-form representations such as (1.2a) and (1.2b) can serve as a check on numerical work.

In fluid problems on wedges, fractional-degree Ferrers functions P_{−1/r},Q_{−1/r} typically appear in the analysis when the wedge angle equals (1−1/*r*)*π*. This includes problems dealing with viscous film coating [32], solidification [33] and vortex layers [34].

### (d) Structure of paper

In §2, the needed properties of the Legendre and Ferrers functions are summarized. Section 3 contains the main results: the just-mentioned collection of two dozen identities, or transformation formulae. In §§4 and 5, it is indicated how the results of §3 are proved. The former section is introductory: it illustrates the method by deriving Whipple's relation and two similar transformations, one new. The proof in §5 employs distinct algebraic curves when *r*=3,4,6. In §6, how to perform integer shifts of the degree *ν* is explained. In §7, explicit formulae are derived, from one of the identities, for

## 2. Basic facts

The (associated) equation of Legendre is the ordinary differential equation
*μ*↦−*μ* and *ν*↦−1−*ν*, as well as under *z*↦−*z*. It can be viewed as an equation on the Riemann sphere, which projects stereographically onto *z*=1,−1 and *μ*/2}, {±*μ*/2} and {−*ν*,*ν*+1}. Hence, locally, any solution of (2.1) is a combination of (*z*−1)^{±μ/2}, (*z*+1)^{±μ/2} and (1/*z*)^{−ν},(1/*z*)^{ν+1}. Or rather, this is the generic behaviour. If the exponents at any singular point differ by an integer, then the local behaviour of the dominant solution, coming from the smaller exponent, will include a logarithmic factor. This causes the familiar logarithmic behaviour as *z*→±1 of *μ*, written as *m*).

By convention, the Legendre functions *μ*>0 and *z*=1, and *e*^{μπi}, *Γ*(*ν*+*μ*+1) and *e*^{μπi} in the definition of *ν*,*μ* and its argument *z*=*x*>1 are all real. (Compare (1.4).)

The advantage of Olver's *z* not on the cut *ν*,*μ*. Also, *μ*/2 exponent at *z*=1 and the *ν*+1 exponent at *μ*≠−1,−2,… and *μ*/2 and −*ν* [35]. The correct asymptotics in these two degenerate cases are given in [1]. It should be noted that there are subcases of the degenerate cases in which *identically zero*. Specifically, if *M* equals 1,2,3,… then *N*=1,2,3,…, then

Many formulae and identities involving *Γ*(*ν*+*μ*+1), and the wish to simplify these formulae partially justifies the introduction of the conventional function *Γ*(*ν*+*μ*+1) factor, *ν*+*μ* is a negative integer, *except* in the just-mentioned sub-case: if *N*=1,2,3,…, then *z*, are not identically zero. Informally, this is because in each of these, the product of *Γ*(*ν*+*μ*+1) (infinite) with

By convention, the Ferrers functions *z*>0, i.e. on the upper and lower half-planes, by

and
*ν*,*μ* for which *μ*≠−1,−2,−3,…) by
*μ*=0, and are related by more complicated integral transforms when *μ*≠0 [36]. This relationship, which is suggested by (2.4), is why

Besides *ad hoc* function is less mysterious than it looks: by [2], 3.3(10), it satisfies

## 3. Main results

The theorems and corollaries numbered 3.1–3.12 contain the main results: a collection of two dozen algebraic Legendre identities, or transformation formulae. Each theorem contains a list of identities in rationally parametrized form, and its corollary gives each identity in a trigonometrically parametrized form, which may be more useful in applications.

The identities come from rational curves *r* equals 3,4 or 6. The following indexing scheme is used. For each *r*, the identities come in pairs. The two pairs coming from *I*_{4}(*i*),*I*_{6}(*i*).

The theorems are ordered, so that the case *r*=4 is covered first, because the curves *r*=6; and finally, *r*=3. The *r*=3 case closely resembles the *r*=6 case, but is deficient in that the order −*α* of the left-hand function must equal zero. The *r*=3 identities given in theorems 3.9, 3.11 and their corollaries cannot readily be generalized to non-zero order, and the same is true of the *r*=4 identities coming from

When the order −*α* is an integer *m*, by applying these identities one can express the Ferrers pair *r* (or by +1/*r*) is explained in §6.

In each identity, the parameter (whether *ξ*, *θ* or *p*) varies over a specified real interval. In fact, each identity extends by analytic continuation to the complex domain, to the largest connected open subset of

### (a) Signature-4 identities

### Definition

The algebraic *L*–*R* curve *R*↦−*R*, which is performed by *p*↦−*p*. An associated prefactor function *A*=*A*(*p*), with limit unity when *p*→1 and (*L*,*R*)→(1,1), is

### Theorem 3.1

*For each pair u, v of Legendre or Ferrers functions listed below, an identity*
*of type I*_{4}*, coming from the curve* *holds for the specified range of values of the parameter p.*

*R*as a function of

*ξ*or

*θ*. This yields the following.

### Corollary 3.2

*The following identities coming from* *hold for* *when* *and* *θ*∈(0,*π*).

### Definition

The algebraic *L*–*R* curve *R*↦4/(*R*+1)−1, which is performed by *p*↦1/*p*. An associated prefactor function *A*=*A*(*p*), with limit unity when *p*→1 and (*L*,*R*)→(1,1), is

### Theorem 3.3

*For each pair u,v of Legendre or Ferrers functions listed below, an identity*
*of type I′*_{4}*, coming from the curve* *, holds for the specified range of values of the parameter p.*

### Remark

*p*−1)/(*p*+1)∈ (0,1) as parameter.

To construct trigonometric versions of these identities, one substitutes *R* as a function of *θ*. This yields the following.

### Corollary 3.4

*The following identities coming from* *hold when* *θ*∈(0,*π*).

### Remark

The right-hand arguments equal

### (b) Signature-6 identities

### Definition

The algebraic *L*–*R* curve *L*↦−*L* and *R*↦−*R*, which are performed by *p*↦3/*p* and *p*↦−*p*. An associated prefactor function *A*=*A*(*p*), with limit unity when *p*→1 and (*L*,*R*)→(1,1), is

### Theorem 3.5

*For each pair u,v of Legendre or Ferrers functions listed below, an identity*
*of type I*_{6}*, coming from the curve* *, holds for the specified range of values of the parameter p.*

*R*as a function of

*ξ*or

*θ*. This yields the following.

### Corollary 3.6

*The following identities coming from* *hold for* *when* *and* *θ*∈(0,*π*).

### Remark

The right-hand arguments equal

### Definition

The algebraic *L*–*R* curve *L*↦−*L* and *R*↦−*R*, which are performed by *p*↦−3/*p* and *p*↦−*p*, and in fact under any transformation of *p* of the Möbius type *p*↦(*Ap*+*B*)/(*Cp*+*D*) that permutes *p*-sphere. (These transformations make up a dihedral group of order 12. Each induces a Möbius transformation of *L*, either *L*↦*L* or *L*↦−*L*, and one of *R*.) An associated prefactor function *A*=*A*(*p*), with limit unity when *p*→1 and (*L*,*R*)→(1,1), is

### Theorem 3.7

*For each pair u,v of Legendre or Ferrers functions listed below, an identity*
*of type I′*_{6}*, coming from the curve* *, holds for the specified range of values of the parameter p.*

### Remark

The factors

The *α*=0 case of *p*−1)/2∈(0,1) as parameter. The case of arbitrary *α* was given in hypergeometric notation by Garvan [38], (2.32).

To construct trigonometric versions of these identities, one substitutes *R* as a function of *θ*. This yields the following.

### Corollary 3.8

*The following identities coming from* *hold for* *when* *θ*∈(0,*π*).

### Remark

The right-hand arguments equal

### (c) Signature-3 identities

### Definition

The algebraic *L*–*R* curve

which is rationally parametrized by
*R*↦−*R*, which is performed by *p*↦−*p*. An associated prefactor function *A*=*A*(*p*), with limit unity when *p*→1 and (*L*,*R*)→(1,1), is

### Theorem 3.9

*For each pair u,v of Legendre or Ferrers functions listed below, an identity*
*of type I*_{3}*, coming from the curve* *holds for the specified range of values of the parameter p.*

*R*as a function of

*ξ*or

*θ*. This yields the following.

### Corollary 3.10

*The following identities coming from* *hold when* *and* *θ*∈(0,*π*).

### Definition

The algebraic *L*–*R* curve *R*↦−*R*, which is performed by *p*↦−*p*. An associated prefactor function *A*=*A*(*p*), with limit unity when *p*→1 and (*L*,*R*)→(1,1), is

### Theorem 3.11

*For each pair u,v of Legendre or Ferrers functions listed below, an identity*
*of type I′*_{3}*, coming from the curve* *holds for the specified range of values of the parameter p.*

### Remark

Identities *p*−1)/2∈(0,1) as parameter.

To construct trigonometric versions of these identities, one substitutes *R* as a function of *θ*. This yields the following.

### Corollary 3.12

*The following identities coming from* *hold when* *θ*∈(0,*π*).

## 4. Additional results

This section presents three additional transformation theorems for Legendre functions, based on rational or algebraic transformations of the independent variable, and introduces the fundamental proof technique. Theorem 4.1 relates the four functions

Theorem 4.1, or an equivalent, has appeared in the setting of ‘generalized’ (associated) Legendre functions; compare [39] and [40], §4. It is not well known. Whipple's formula is more familiar, in part because it has two free parameters rather than one, but the proof indicated below is new. Theorem 4.5 relates the functions

The calculations in the proofs employ the calculus of Riemann P-symbols, which is classical [41]. For any homogeneous second-order ordinary differential equation *μ*/2 at *z*=1, the exponent *μ*/2 at *z*=−1, the exponent *μ*/2 at *z*=1 and the exponent *ν*+1 at

Changes of variable applied to an equation *w*(*z*)=(*z*−*z*_{0})^{c}*u*(*z*) is a linear change of the dependent variable, the transformed equation *z*=*z*_{0} shifted upward by *c* relative to those of

Any rational map *f* is a homography (also called a linear fractional or Möbius transformation), so that *AD*−*BC*≠0. In this case, *f* provides a one-to-one correspondence between the points of the *z*-sphere and those of the

Also, if *f* is a rational function with *f*^{−1}(*z*_{0}) equals *k*, the exponents at the lifted point *k* times those at *z*=*z*_{0}. (This assumes that the points

The calculus of P-symbols is a powerful tool for exploring the effect of changes of variable on Fuchsian differential equations, but a P-symbol does not, in general, uniquely determine such an equation, or even its solution space. If a second-order equation has *m* specified singular points on the Riemann sphere (*m*≥3), it and its two-dimensional solution space are determined by the 2*m* exponents and by *m*−3 *accessory parameters* [41]. To prove equality between two second-order differential equations with more than three singular points, which have the same singular point locations and characteristic exponents but have been obtained by different liftings, one must work out the lifted equations explicitly, and compare them term-by-term.

### (a) Homographic identities

Theorem 4.1 is based on an algebraic change of variable, from *L* to *R*, which is relatively simple: it is a homography of the Riemann sphere. The associated curve is denoted *M*.

### Definition

The algebraic *L*–*R* curve *p*↦1/*p*. An associated prefactor function *A*=*A*(*p*), equal to unity when *p*=1 and (*L*,*R*)=(1,1), is

### Theorem 4.1

*For each pair u,v of Legendre or Ferrers functions listed below, an identity*
*of type M, coming from the curve* *holds for the specified range of values of the parameter p.*

*R*as a function of

*ξ*or

*θ*. This yields the following.

### Corollary 4.2

*The following identities coming from* *hold for* *when* *and* *θ*∈(0,*π*).

### Remark

Owing to the invariance under *α* is a half-odd-integer, these identities become singular. If

### Proof of theorem 4.1.

Functions *u*,*v* satisfy Legendre's equation (2.1), of degree *ν*=*α*−1/2 and order *μ*=−2*α*, if and only if *u*(*L*(*p*)) and *A*(*p*)*v*(*R*(*p*)) both satisfy a certain second-order differential equation with independent variable *p*, which is obtained by lifting. This can be checked by a P-symbol calculation, noting that the inverse images of the singular points *L*=*L*(*p*), and *R*=*R*(*p*). The left and right P-symbols are
*L*↦*R* homography, followed by a linear change of the dependent variable coming from the prefactor

It remains to show that for each *u*,*v* listed in the theorem, the functions *u*(*L*(*p*)) and *A*(*p*)*v*(*R*(*p*)) are the *same* element of the two-dimensional solution space of the lifted equation. First, note that this is true up to some constant factor, because each of *u*,*v* is a Frobenius solution associated with a singular point and one of its exponents, and the points and exponents correspond. In identity *α*+1/2, and *R*=−1 with exponent *α*. Examining the above P-symbols reveals that for *u*(*L*(*p*)) and *A*(*p*)*v*(*R*(*p*)), which are Frobenius solutions of the lifted equation, are associated with its singular point *α*+1/2; so they must be constant multiples of each other.

Finally, one must check that for each identity, the constant of proportionality equals unity. This follows by comparing the left- and right-hand sides near the common singular point. Identity

### (b) Whipple's formula

Theorem 4.3 is a version of Whipple's *L* to *R* was introduced by Whipple, the underlying curve is denoted

### Definition

The algebraic *L*–*R* curve

and is invariant under *L*↦−*L* and *R*↦−*R*, which are performed by *p*↦(*p*+1)/(*p*−1), 1/*p* and −*p*, and under the group they generate (which can be viewed as a dihedral group of order 8, acting on *A*=*A*(*p*), equal to unity when *L*=*R*, is

### Theorem 4.3

*For each pair u,v of Legendre functions listed below, an identity*
*of type W*_{2}*, coming from the curve* *holds for the specified range of values of the parameter p.*

*R*as a function of

*ξ*; it equals

**rather than**

*Q*

### Corollary 4.4

*The following identities coming from* *hold for* *when*

### Proof of theorem 4.3.

Similar to the proof of theorem 4.1, this follows from lifting Legendre's equation (2.1), now of degree *ν*=*α*−1/2 and order *μ*=−*β*, to the Riemann *p*-sphere, along the covering maps *L*=*L*(*p*) and *R*=*R*(*p*). (The latter lifting is followed by a linear change of the dependent variable coming from the prefactor *L* or *R* is the subset *p*-sphere, and the left and right P-symbols both turn out to be
*L*^{−1}(1),*L*^{−1}(−1),*R*^{−1}(1),*R*^{−1}(−1), which, respectively, equal

The identities of the theorem, each based on a pair (*u*,*v*), come from the table
*p*-line is divided by the singular points, is mapped monotonically onto a real *L*-interval and a real *R*-interval. (An asterisk indicates a change of direction.) For each *p*-interval, the possible (*u*,*v*) are determined thus: if each of *u* and *v* is to be one of *u*(*L*) and *v*(*R*) (namely 1 for *P* and P, −1 for *u*(*L*(*p*)) and *A*(*p*)*v*(*R*(*p*)) will be the same Frobenius solution in the (two-dimensional) solution space of

On the real axis, *p*-interval in (4.10) other than *L* or *R* ranges between −1 and *u*,*v*), namely *W*_{2}(*i*) and *p*=1 for *W*_{2}(*i*) and

### (c) Whipple-like relations

Theorem 4.5 contains an unexpected pair of Whipple-like identities, which are based on an algebraic curve of higher degree. Because its parametrization resembles that of the Whipple curve *W*_{4}.

### Definition

The algebraic *L*–*R* curve *L*↦−*L* and *R*↦−*R*, which are performed by *p*↦(*p*+1)/(*p*−1), 1/*p* and −*p*, and under the group they generate (the same as for *A*=*A*(*p*), equal to unity when *L*=*R*, is

### Theorem 4.5

*For each pair u,v of Legendre functions listed below, an identity*
*of type W*_{4}*, coming from the curve* *holds for the specified range of values of the parameter p.*

*R*as a function of

*ξ*. This yields the following, which owing to the

### Corollary 4.6

*The following identities coming from* *hold for* *when*

### Proof of theorem 4.5.

This closely resembles the proof of theorem 4.3. The inverse image of the set of singular points *L* or *R* is now the subset *p*-sphere, and the left and right P-symbols both turn out to be
*α* were replaced by −*β*, as in theorem 4.3, the two P-symbols would differ; which is why theorem 4.5 has only one free parameter, namely *α*.) As before, the lifted equations *p*-intervals *L*-interval and a real *R*-interval, is valid without change. ▪

### Remark

One may wonder how the curve *p*↦*L*(*p*),*R*(*p*). In fact, the Whipple-like identities of type *W*_{4} were conjectured first, and the curve was engineered to provide a proof. As the reader can verify, they follow from homographic identities (of type *M*, above) by applying Whipple's formula to both sides.

Many (associated) Legendre functions of half-odd-integer degree and order, including

Durand [18] has interpreted Whipple's relation in quantum–mechanical terms, as an automorphism of a conformal Lie algebra that extends the more familiar algebra *so*(2,1). It is unclear whether the identities of type *W*_{4} have a similar interpretation.

## 5. Derivation of main results

The two dozen identities in the theorems and corollaries numbered 3.1–3.12, arising from algebraic curves *r*=3,4,6, are proved by the technique developed in §4. The key fact is that in the four identities of each theorem, the left and right functions *u*(*L*(*p*)), *A*(*p*)*v*(*R*(*p*)) satisfy the same differential equation, as functions of the parametrizing variable *p*. This equality (i.e. *p*-sphere, for full rigour, they must be worked out explicitly, and compared.

Once *p*↦*L*(*p*),*R*(*p*) determine the associated identities: in particular, which of the Legendre functions *u*,*v*. The algorithm for finding the possible *u*,*v* was illustrated in §4. For each real *p*-interval delimited by real singular points, one checks whether the *L*-range or *R*-range is *p*-interval is rejected. An *L*-range or *R*-range that is (−1,1), or a subset of it, corresponds to a Ferrers function, and similarly, *P* and P, −1 for *p*-interval. Any constant of proportionality needed between the two sides is calculated by considering their asymptotic behaviour at this singular point (see §2).

The preceding algorithm suffices to derive or verify all the identities of §3, except for *r*=3,4,6. Anomalously, these relate *p*-intervals to *L*,*R*-intervals) should suffice for the interested reader to confirm all identities other than these. It is exponent *differences* that are supplied below, because unlike exponent pairs they are unaffected by the replacement of *v*(*R*(*p*)) by *A*(*p*)*v*(*R*(*p*)).

### (a) Signature-4 identities

On the curve *p*-sphere, the equation *p*=−1,0,1. The respective exponent differences are *α*,2*α*,*α*. It also has a ‘removable’ singular point at *p*-intervals and monotonic *p*↦*L*,*R* maps are tabulated as
*p*-interval *R*>0, its left-hand and right-hand side functions are *P*,P. This is identity *I*_{4}(*i*) of theorem 3.1. The defining singular points of *P*,P (respectively *L*=1, *R*=1) are at *p*=1, i.e. are at the same end, and the prefactor 2^{α} in the theorem comes from requiring the two sides to agree at *p*=1. In the same way, the *p*-interval (0,1) yields both *I*_{4}(*ii*) and *P* and *p*=1 and *p*=0. The *p*-interval *I*_{4}(*i*) by *R*↦−*R*, which is performed by *p*↦−*p*.

On the curve *p*=−1. The equation is
*α*, as in the *I*_{4} identities, then *α*=0; which is why theorem 3.3 includes no free *α* parameter.) The real *p*-intervals and monotonic *p*↦*L*,*R* maps are tabulated as
*p*-interval *I*_{4}′(*i*) and *p*-interval (0,1) yields both *I*_{4}′(*ii*) and *P* and

### (b) Signature-6 identities

On the curve *p*-sphere, the equation *α*,*α*,4*α*,*α*,*α*,4*α*. It also has ‘apparent’ singular points at *p*-intervals and monotonic *p*↦*L*,*R* maps are tabulated as
*p*-interval *I*_{6}(*i*) of theorem 3.5, relating *P*,P, and the *p*-interval (0,1) yields both *I*_{6}(*ii*) and *P* and *p*-intervals *L*↦−*L* and *R*↦−*R*, which are performed by *p*↦3/*p* and *p*↦−*p*.

On the curve *α*,2*α*,2*α*,2*α*,2*α*,2*α*. It also has apparent singular points at *p*-intervals and monotonic *p*↦*L*,*R* maps are tabulated as
*p*-interval (1,3) yields identities *I*_{6}′(*i*) and *p*-interval (0,1) yields both *I*_{6}′(*ii*) and *P* and *p*-intervals (−3,−1) and *L*↦−*L* and *R*↦−*R*, which are performed by *p*↦−3/*p* and *p*↦−*p*.

### (c) Signature-3 identities

On the curve *p*-sphere, the equation *α*.) The real *p*-intervals and monotonic *p*↦*L*,*R* maps are tabulated as
*p*-interval *I*_{3}(*i*), relating *P*,P, and the *p*-interval (0,1) yields the pair *I*_{3}(*ii*) and *P* and *p*-interval *I*_{3}(*i*) by *R*↦−*R*, which is performed by *p*↦−*p*. The *p*-interval

On the curve *α*.) The table of real *p*-intervals and monotone *p*↦*L*,*R* maps is the same as for

### (d) Finer asymptotics

It has now been explained how each identity in the theorems in §3 is derived, except for *r*=3,4,6. Each of these relates a *ad hoc* Legendre function on the left (a linear combination of *R*-interval, over which the Ferrers argument ranges, does not extend the entire way from *R*=1 to *R*=−1. This is why the above proof technique, applied to this *R*-range and the corresponding *p*-interval, produced only one identity (i.e. *I*_{r}(*i*)), which came by requiring identical left and right asymptotics at the *R*=1 end: at the singular point *p*=1. The local behaviour at the other end, which is not a singular point, is not given by any simple formula.

This difficulty can be worked around by focusing on the *p*=1 end of the relevant *p*-interval (which is *r*=4,6,3), but employing finer asymptotic approximations. The leading behaviours of *z*→1 are given in (2.2a), (2.5). Those of *z*→1 are more difficult to compute. (The point *z*=1 is not the defining singular point for *μ* is not an integer and *ν*±*μ* are not negative integers,
*z*−1 replaced by *z*, if the first term on the right-hand side is multiplied by *μ*/2,+*μ*/2.

It is easily checked that if in theorems 3.1, 3.5 and 3.9, the right function *v* equals 2/*π* times the specified Ferrers function Q, and the left function *u* equals *fine* asymptotics at *p*=1: the coefficients of each of the two Frobenius solutions will be in agreement. In fact, it was to obtain this agreement that the *ad hoc* Legendre function

In deriving identity _{−1/2}, with both functions of order zero (there is no *α* parameter). In the asymptotic development of *z*→1, the Frobenius solutions (*z*−1)^{−μ/2},(*z*−1)^{μ/2} of (5.12) are replaced by

## 6. Elliptic integral representations

The now-proved identities of section 3, joined with differential recurrences for Legendre and Ferrers functions, lead to useful representations in terms of the first and second complete elliptic integrals, *K*=*K*(*m*) and *E*=*E*(*m*), the argument *m* denoting the elliptic modular parameter.

### Theorem 6.1

*The Legendre functions* *and Ferrers functions* *,* *, where the degree ν differs by ±1/r (r=2,3,4,6) from an integer and the order m is an integer, can be expressed in closed form in terms of the complete elliptic integrals K,E.*

### Proof.

The case *r*=2 is well known (the Legendre functions of half-odd-integer degree and integer order are the classical toroidal functions). The fundamental representations are
*ν*,*m* are handled by standard differential recurrences on the degree and order.

Let *M*^{±} and M^{±}, are defined (with *D*_{ξ}=*d*/*dξ* and *D*_{θ}=*d*/*dθ*) by
*C*^{−} equals *C*^{+} equals unity, and the sign factor *s* has the following meaning: *s*=1,−1 for *M*_{±} and M_{±}, are given by

By applying these recurrences to any of *ν* is a half-odd-integer, one can express it in terms of the corresponding *K*=*K*(*m*) and *E*=*E*(*m*), which are

The preceding algorithm is easily extended from *r*=2 (the classical case) to *r*=3,4,6. Suppose one were given a Legendre function *r* identities *r*=4,6,3 respectively, will express these Legendre functions in terms of the Ferrers functions P_{−1/2},Q_{−1/2}. (Recall that *I*_{r}(*i*), is a linear combination of *r*=3,4,6 reduce to the classical case.

If one were given a Ferrers function, one of *r* identities (say, the pair

The only thing that remains to be explained is how to handle the case when the degree *ν* differs by +1/*r* rather than −1/*r* from an integer. The additional effort required is minor. The functions *ν* by −*ν*−1, which interchanges the two cases; and as for

The algorithm in this proof is not optimal when *r*=4,6. For these two values of *r*, the Ferrers functions *r*=3 if *m* is non-zero. This enhancement for *r*=4,6 may be of numerical relevance, because the recurrences for Legendre and Ferrers functions are often numerically unstable, and modern schemes for evaluating toroidal functions do not employ them [12].

## 7. Algebraic Legendre functions

One of the identities of §3, the signature-6 identity *r*=3,4,6) that were covered in the last section. No elliptic integrals are involved.

transforms *μ* is a half-odd-integer. This involves only elementary functions [1], §14.5(iii). In fact, any Legendre or Ferrers function with (i)

### Theorem 7.1

*The following formulae hold when θ∈(0,π) and* *.*
*Moreover,*
*where* *is the prefactor.*

### Remark

To obtain explicit formulae when the degree and order differ by integers from those appearing in this theorem, one would apply differential recurrences, as in the last section.

### Proof.

An explicit formula for *α*=1/4 case of *θ* being replaced by *θ*+*π*, becomes the one for

The formula for *θ*=*i* *ξ*). The one for *C*, then comes by applying Whipple's transformation. The equality of the two values given for *C* comes from a gamma-function identity [43], p. 270. ▪

The formulae of theorem 7.1 can be written in algebraic form, because *z*, expressible in terms of radicals, is the following. Legendre's differential equation (2.1) on the Riemann sphere *z*=±1 and *μ*,*μ*,2*ν*+1. It is a classical result of Schwarz (see [2], § 2.7.2, [41], ch. VII and [44]) that for a differential equation of the hypergeometric sort to have *only algebraic solutions*, its unordered triple of exponent differences must be one of 15 types, traditionally numbered I–XV. The case when (*ν*,*μ*)=(−1/4,−1/3) and (*μ*,*μ*,2*ν*+1)=(−1/3,−1/3,1/2) is of Schwarz's type II, and the case when (*ν*,*μ*)=(−1/6,−1/4) and (*μ*,*μ*,2*ν*+1)=(−1/4,−1/4,2/3) is of type V.

For each type in Schwarz's list, there is a (projective) monodromy group: the group of permutations of the branches of an algebraic solution that is generated by loops around the three singular points. (Strictly speaking, the algebraic function referred to here is not a solution *u*(*z*) of the equation, but rather the ratio *u*_{1}(*z*)/*u*_{2}(*z*) of any independent pair of solutions; which is the import of the term ‘projective.’) For Schwarz's types II and V, the respective groups are tetrahedral and octahedral: they are isomorphic to the symmetry groups of the tetrahedron and octahedron, which are of orders 12 and 24. It is no accident that as an algebraic function of

An interesting consequence of the formula for _{2}*F*_{1}. Taking into account the relation
*x*<0, and in fact on the complex *x*-plane with cut *x*; provided, that is, that the branch of each radical is appropriately chosen.

It has long been known how to obtain *parametric* formulae for algebraic hypergeometric functions [41], ch. VII, and such a formula for

## 8. A curiosity

Until this point, each Legendre transformation formula derived in this paper has been related at least loosely to the transformations in Ramanujan's theory of signature-*r* elliptic integrals. More exotic Legendre transformations exist, as the curious theorem and corollary below reveal. They relate

### Definition

The algebraic *L*–*R* curve *L*,*R*)↦(−*L*,−*R*), which is performed by *p*↦−1/*p*. An associated prefactor function *A*=*A*(*p*), equal to unity when *p*=0 and (*L*,*R*)=(1,1), is

### Theorem 8.1

*For each pair u,v of Legendre or Ferrers functions listed below, an identity*
*of type X, coming from the curve* *, holds for the specified range of values of the parameter p.*

To construct trigonometric versions of these identities, one substitutes *X*(*i*) and *X*(*ii*) respectively, and then writes *R* in terms of *θ*. This yields the following.

### Corollary 8.2

*The following identities coming from* *hold when* *and* *respectively*.
*In both, the constant prefactor* *C* *equals* *Γ*(6/5)/2*Γ*(11/10).

### Proof of theorem 8.1.

This resembles the proofs in §5 of the main results. On *p*-sphere, the equation *A*(*p*)*v*(*R*(*p*)) both take the form
*p*-intervals and monotonic *p*↦*L*,*R* maps are tabulated as
*p*-interval *X*(*i*), relating P,P, and the *p*-interval *X*(*ii*), relating P,*P*. The *p*-interval *X*(*i*) by (*L*,*R*)↦−(*L*,*R*), which is performed by *p*↦−1/*p*.

From the covering map *R*=*R*(*p*) of (8.2) and the table (8.4), one would expect that *p*=−2 (in *R*^{−1}(1)), *R*^{−1}(−1)) and

Because the cardinality of the set of singular points

The rather mysterious algebraic curve *X*(*i*),*X*(*ii*), their validity is in some way linked to the equations 3^{2}+4^{2}=5^{2} and 2^{2}+11^{2}=5^{3}. The existence of other exotic algebraic curves that lead to Legendre or Ferrers identities will be explored elsewhere.

## 9. Summary and conclusions

It has been shown that Legendre and Ferrers functions of fractional degree (*r*=3,4,6 as well as *r*=2) can be expressed in terms of complete elliptic integrals. This is made possible by the identities exhibited in section 3, in their rationally and trigonometrically parametrized forms. There are many applications of these identities in the mathematical and physical sciences, as was indicated in the Introduction. One interesting application is to the Fourier cosine series for the periodic function *x*<1. As was shown, each of its coefficients can be expressed symbolically in terms of complete elliptic integrals.

The collection of identities in the theorems and corollaries 3.1–3.12 can be considerably expanded by composing identities together. For instance, any Legendre function of the second kind (*K*=*K*(*m*), to which it reduces if *α*=0. Many other examples could be given.

The algebraic curves, *r*=3,4,6, were presented in such a way as to seem natural tools for specifying algebraic transformations of arguments of Legendre functions, i.e. *I*_{r}′ coming from *I*_{6}′(*ii*),*I*_{6}(*ii*) differ by, respectively, containing *I*_{6}′(*ii*) and the latter relation were found first, and the identity *I*_{6}(*ii*) and the curve

The signature-4 identities of types *I*_{4},*I*_{4}′ are of a known kind, because their *L*=*L*(*R*). When written in terms of the Gauss function _{2}*F*_{1}, they become quadratic hypergeometric transformations. But the identities of types *I*_{r},*I*_{r}′, *r*=3,6 are based on

## Competing interests

I have no competing interests.

## Funding

I received no external funding for this work.

- Received February 8, 2016.
- Accepted March 29, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.