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Fractional operators and multi-integral representations for associated Legendre functions

In a recent paper, Cohl and Costas-Santos derived a number of interesting multi-derivative and multi-integral relations for associated Legendre and Ferrers functions in which the orders of those functions are changed in integral steps. These are of potential use in a number of physical problems. We show here how their results can be derived simply from more general relations involving non-integer changes in the order obtained using the fractional group operator methods developed earlier for SO(2,1), E(2,1) and its conformal extension, and SO(3). We also present general integral relations for fractional changes of the degrees of the functions, and related multi-derivative and multi-integral representations.


I. INTRODUCTION
In a recent paper [1], Cohl and Costas-Santos derived several multi-derivative and multi-integral representations for general Legendre functions and Ferrers functions (the associated Legendre functions "on the cut") which change the order of those functions in integral steps. They used the results in further investigations, obtaining a number of new results on those functions and their integrals. Their basic results are of considerable potential interest in a number of physical problems, as summarized in some detail in that paper.
In the present paper, we present alternative derivations of the basic results in [1], and show that they can be obtained quite easily using the fractional operator methods developed for Bessel functions and associated Legendre functions in [2] and [3]. The results in [1] appear here as limiting cases of more general expressions. We also present similar relations not considered by Cohl and Costas-Santos in which the degrees of the Legendre functions are changed.
The Legendre functions appear naturally in the representation theory for the Lie groups SO(3) and SO(2,1) considered as the groups of spherical and hyperbolic transformations on the sphere S 2 and the hyperbola H 2 , and on the conformal extension of SO(2,1) to E(2,1). See, for example, the extensive discussions in [4] and [5]. The Lie algebras of these groups are realized through the action of linear differential operators D(w, ∂ w ) on representation functions F α (w) of several variables w, with the multi-labels α labeling the realization of the Lie algebra. When reduced to a single variable, the actions of appropriate elements D-raising and lowering operators-are schematically of the form DF α = cF α±1 , and lead to the standard differential recurrence relations for the Legendre functions.
The exponentials e −tD , defined by Taylor series expansions in the group parameter t, are elements of the Lie group. Their action on the functions F α (w) can be interpreted in terms of transformations of the functions through transformations of the coordinates w on S 2 or H 2 , and lead to many sum rules, generating functions, and integral representations for the F s; see, e.g. [2][3][4].
As in [2] and [3], we will assume that the group action e −tD F can be defined for all t, and will define Weyl-type fractional operators D λ W as integrals over group elements, where the contour C W = (∞, 0 + , ∞) in the complex t plane runs in from ∞, circles the origin in the positive sense, and runs back to ∞. Both the direction of the contour and the functions F must be chosen to obtain convergence of the integral. A similar Riemann-type fractional operator is given by D λ R (w, ∂ w ) = 1 2πi e iπλ Γ(λ + 1) CR dt t λ+1 e −tD(w,∂w) F (w), (2) where the contour C R = (w 0 , 0 + , w 0 ) again circles the origin in the positive sense, but ends at a finite point t = w 0 where the potential endpoint contributions to the integral vanish. The results of the Weyl-and Riemann-type expressions can be related in many instances, including the cases of the reduced single-variable stepping operators used in later sections, to Weyl and Riemann fractional integrals ( [6], Chap. 13), hence the names. As shown in [2], the fractional operators have the expected algebraic properties: The inverse of D λ is just D −λ where 1 is the unit operator. We used the results on fractional group operators in [2,3] to obtain a large number of generating functions and integral representations for Bessel and associated Legendre functions, many new, including double integral representations for the associated Legendre functions which change both the order and the degree of those functions by different non-integral steps. We can also use the fractional operators to extend or reinterpret standard results; an example is given in the Appendix where we show that the Rodrigues formula for the Jacobi polynomials P (α,β) n (z) holds for non-integer as well as integer n.
The use of the present methods to obtain known and new results also puts those in a more general group-theoretic setting than is encountered, for example, in [4,7,8], where the special functions appear as the basis functions for unitary representations of the relevant groups. The unitarity condition restricts the values of the parameters that can appear, hence the generality of the results obtained. The special functions appear here only as functions upon which one can realize the actions of the relevant Lie algebras, with no further restrictions.
In the following sections, we will use fractional group operators in SO(2,1) and E(2,1) to reinterpret and extend the results of Cohl and Costas-Santos [1]. We will first briefly review the properties of the operators in SO(2,1) and E(2,1) needed in our constructions, and then extend the results on fractional operators and Legendre functions obtained in [3]. We will deal with four operators, M + and M − , and K 3 and P 3 . The operators M ± raise and lower the order µ of a Legendre function F µ ν ; K 3 and P 3 raise and lower the degree ν. We will consider Weyl and Riemann-type relations for each, and determine how the relations apply to general functions in the complex plane and to the functions on the cut −1 ≤ x ≤ 1, the Ferrers functions.
The basic results of Cohl and Costas-Santos [1] will appear in this framework simply as expressions of the form for raising operators D + , and for lowering operators D − , where the first set on each line involves n-fold derivatives, and the second (inverse) set, n-fold integrations, n integer. The general results with respect to the change of degree do not appear in [1]. The content of the paper is as follows. In Sec. II, we review the basic properties of the operators M ± , K 3 , and P 3 , including their algebraic properties and relation to the underlying group structure of SO(3,1) and E(2,1), and show how their action on the associated Legendre function P µ ν (z) and Q µ ν (z) can be reduced to that of simple derivatives. Our analysis differs in this respect from that in [3], where we used the full form of the operators, with concomitant complication. In Sec. III, we examine the change of the order µ of the Legendre functions using the fractional stepping operators M λ ± . We begin in Sec. IIIA by considering the effect of Weyl-type operators M λ + on Q µ ν (z). We then continue to the case of P µ ν (z) where the action is more complicated and yields not only the expected function P µ+λ ν (z), but also an extra term proportional to Q µ+λ ν (z). This demonstrates that the underlying algebraic structure does not completely determine the action of the fractional operators; analytic checks are also necessary. We further show how the results for both Q µ ν (z) and P µ ν (z) can be rewritten as the multi-derivative and multi-integral expressions derived in [1] for λ = ±n integer.
In Sec. IIIB, we consider the Weyl-type fractional lowering operators M λ − . The definition of these operators involves a rotation of the initial integration contour appropriately away from the singularities of the integrand to obtain properly defined operators. The results we obtain for for Q µ ν and P µ ν are apparently new, and can be converted to new Weyltype fractional integrals. In constrast to the case of M λ + , the action of M λ − on P µ ν does not introduce an extra function of the second kind in the result, but the coefficient and phase of the final function are different from those obtained for Q µ ν ; this is again not evident from the underlying algebraic expressions. For λ = ±n integer, the results for both Q µ ν and P µ ν can be reduced to the multi-derivative and multi-integral expressions derived in [1]. In Sec. IIIC, we consider the Riemann-type fractional operators M λ ± . Given the finite z-dependent endpoints of the integration contours discussed there, it is necessary to check explicitly that the action of those operators on solutions of the associated Legendre equation again gives solutions of that equation for the appropriately modified order. That is the case for the action of M λ + on the Legendre functions P µ ν ; the result gives a new derivation and extension of a known fractional integral. In contrast, the action of the Riemann version of M λ + on functions Q µ ν leads to functions that satisfy an inhomogeneous version of the Legendre equation, and can be expressed in terms of hypergeometric functions of type 3 F 2 , again not evident algebraically. There are no proper Riemann-type expressions M λ − for its action on Legendre functions of either type. We consider in detail the action of that operator on P µ ν , a case considered for λ integer by Cohl and Costas-Santos [1]; the general result can again be expressed as a 3 F 2 , and its behavior analyzed in detail using a Barnes-type representation for that function. The results obtained in this section can again be reduced to multi-derivative and multi-integral expressions as in [1].
In Sec. IV, we consider the change of the degree of the associated Legendre functions using the fractional degreeraising operator K λ 3 and the degree-lowering operator P λ 3 . This involves an added element: an inner automorphism of the complete operator algebra discussed in Sec. II relates these operators to M λ − and M λ + , respectively. This is implemented through Whipple transformations of the Legendre functions. We obtain Weyl-type relations for the action of K λ 3 , but no Riemann-type relations, and both Weyl-and Riemann-type relations for P λ 3 . These results are new. We also obtain corrsponding multi-derivative and multi-integral relations not considered in [1].
Finally, in Sec. V, we briefly consider the case of SO(3) and the Ferrers functions P µ ν (x) and Q µ ν (x) (the Legendre functions on the cut −1 < x < 1). The only natural fractional operators in this case are of the Riemman type, and only exist for L λ + acting on the functions P µ ν . The corresponding results for the functions Q µ ν satisfy an inhomogeneous version of the associated Legendre equation. We generalize the results of Cohl an Costas-Santos for the action of L λ + on functions of the second kind, and again obtain their multi-derivative and multi-integral relations for λ = ±n.
Riemann-type transforms where the contours are finite, the integrals may converge for some input functions, but have nonvanishing endpoint contributions. In those cases, the transform integrals satisfy inhomogeneous rather than homogeneous versions of the Legendre equation.
It also follows from the commutation relations [iM 3 , M ± ] = ±M ± and the definitions of the fractional operators M λ ± in Eqs. (1) and (2) that so that M λ + increases and M λ − decreases the order µ of the Legendre function by λ ( [3], Sec. VI.). However, these two operators do not commute (Eq. (10)), act naturally on different functions (Eqs. (32)-(35)), and are not inverses of each other.
It is important to recognize that the final functions M λ ± F µ ν may not involve the same combination of the Legendre functions P µ ν and Q µ ν as F µ ν ; it is only determined algebraically that they again are solutions of the associated Legendre equation of the same degree ν and of orders µ ± λ. The output combination and normalization can be checked using the different behaviors of P µ ν (z) and Q µ ν (z) for z → ∞ and z → 1 and the relations between the two types of functions. These considerations are not changed by the change to the modified functions f µ ν in Eqs. (17)-(21), and we will use those functions and the reduced forms of the operators in what follows.
A. Weyl-type relations for M λ + 1. Relations for Q µ ν As discussed earlier, Weyl-type fractional operators are defined by integrals on infinite contours, with the integrands vanishing sufficiently rapidly that there are no contributions from the end regions. In the case of the reduced forms of the operators we use here, they are closely related to Weyl fractional integrals ( [6], Sec. 13.2), hence our nomenclature. Using the definition of the Weyl-type operators in Eq. (1) with the reduced form of M + , and choosing z 2 − 1 −µ/2 Q µ ν as the initial Legendre function, we find that The singularities of the integrand at t = −(z ± 1) are to the left of the contour for ℜz > 0, and recede toward −∞ for z → ∞. We can therefore determine the asymptotic behavior of the integral for z → ∞ by letting t = zu, and scaling out z using the limiting form of the integrand for z large. The remaining contour integral gives a multiple of the analytic continuation of a standard integral representation for the beta function ( [9], Eq. 1.5(2); [10], Eq. 5.12.11), with x = −λ and y = ν + µ + λ + 1. The two sides of Eq. (42) have the same limiting behavior z −ν−µ−λ−1 for z → ∞ and equal coefficients, establishing that the integral indeed gives z 2 − 1 −(µ+λ)/2 Q µ+λ ν (z), with the indicated coefficient e −iπλ in Eq. (42) corresponding to the expected factor (−1) λ from the minus sign on the right-hand side of Eq. (33).
It is convenient to introduce an extra factor e −iπµ on both sides of the equation, giving The phases on the two sides of the equation then just eliminate the phases e iπµ included in the conventional definition of the functions Q µ ν (z) in Eq. (15). We will now assume for definiteness that ℜλ > 0, so that the operation M λ + increases the real part of µ. The inverse operation M −λ + then decreases ℜµ, with ℜ(ν + µ − λ + 1) > 0. We obtain the expression in Eq. (46) by collapsing the initial contour and using the reflection formula for the gamma function, Γ(−λ + 1)Γ(λ) = π/ sin(πλ) ( [10], Eq. 5.5.3) to reduce the result. The substitution v = z + t converts that expression to a known Weyl fractional integral ( [6], Eq. 13.2(30)). For λ = n a positive integer, Eqs. (42) and (46) reproduce the results in Remark 5 and Thm. 3 of Cohl and Costas-Santos [1]. In particular, for n integer, M n + = (−1) n (d/dz) n . The contour in Eq. (42) can be closed around t = 0 and the integral evaluated directly using the Cauchy residue theorem. The result after eliminating an overall factor of (−1) n is This is just an n-fold iteration of the derivative relation for M + in Eq. (33).
In the case of the inverse operator M −λ ℜ(ν + µ − n + 1) > 0. This can be rewritten using shifted variables u 1 = z + t 1 , u 2 = u 1 + t 2 , . . . , u n = u n−1 + t n as a result equivalent to Thm. 3 in [1]. We can obtain the same result by repeated partial integrations of the general expression in Eq. (46) with λ = n. As we remarked in the Introduction, this structure is to be expected. M n + is a derivative operator. Its inverse, M −n + , should therefore involve integration as the inverse of differentiation, schematically M n + = (−d/dz) n so M −n + = ( ) n . More generally, for ℜλ > 0, M λ + is a fractional derivative, M −λ + , a fractional integral. We note, as an example of the derivative relations and their inverses, that the particular cases µ = 0 and µ = n in Eqs. (47) and (49), give the simple expressions The case of a Weyl-type integral for the for the action of M λ To proceed, we will write P µ ν (z) in terms of functions of the second kind as ( [9], Eq. 3.3.1(3); [10], Eq. 14.9.12) We have established the action of M λ + on Q µ ν (z) for general ν in Eq. (42). The result there can be used for both terms in Eq. (55) provided ℜ(−ν + µ + λ) > 0, where we assume that ℜν > − 1 2 . This gives (56) The sine factors in Eq. (56) still only involve µ, and not the shifted order µ + λ. The result therefore cannot be expressed simply as a multiple of P µ+λ ν (z), but includes an extra term proportional to Q µ+λ In the case λ = n integer, the second term in Eq. (57) vanishes, and with Closing the contour for n > 0, we find that in agreement with the n th iteration of Eqs. (32) and (33) for F µ ν = P µ ν . The inverse operation M −n + with n > 0 again gives a multiple integral, with This expression can be derived by rewriting the result in Eq. (59) as an n-fold integral, and then following the procedure used for M −n The general Weyl-type expression for M λ − using the functions in Eq. (34) is where the direction of approach of the contour to infinity must be chosen appropriately.
To define the phases, we will assume that the contour runs above the singularities of the integrand at t = z ± 1 which we will take just below the real axis. We will further assume for simplicity that ℜλ < 0 so that the contour in Eq. (62) can be collapsed, giving for the case of F µ ν = Q µ ν , namely, The singularities of the integrand are close to the axis and remain so for z → ∞ parallel to the axis, preventing the use of the asymptotic form of the integrand to estimate the integral.
To move the path of integration away from the singularities, we rotate the contour clockwise by π so that t → e iπ t ′ , 0 ≤ t ′ < ∞. Changing to t ′ as the integration variable introduces an overall factor e −iπλ . Dropping the prime gives the expression valid for ℜ(ν − µ + λ + 1) > 0. In this form, the singularities recede to −∞, away from the integration contour, for z → +∞. As a result, we can easily determine the asymptotic limit of the integral by letting t = zu, scaling out z using the asymptotic form of Q µ ν (z(u + 1)) for z → ∞, and evaluating the remaining integral in terms of a beta function. The limiting behavior is that of a multiple of . The result, after re-instating the integration on the contour (∞, 0+, ∞) to allow continuation to ℜλ > 0, is For λ = n a positive integer, we can reduce the contour in Eq. (65) to a loop around the origin, and the Cauchy residue theorem then gives the relation d dz the expected n th iterate of the relation in Eq. (35). If we write the inverse operation M −n − as an n-fold product M −1 − · · · M −1 − , a calculation similar to that which led to Eq. (49) gives the multiple integral representation Cohl and Costas-Santos [1] do not consider results analogous to the actions of M − on the functions Q µ ν (z) in the complex plane. The results in this subsection, including the multi-integral expression in Eq. (68), are therefore new.
Using the expressions for P µ ν (z) in terms of functions of the second kind in Eq. (55), and for the action of M λ − on those functions in Eq. (66), we find that There is no extra term proportional to Q µ−λ ν . However, the overall coefficient on the right-hand side of the equation (66), and there is a change in phase connected to the phase in Eq. (55).
For ℜλ > 0, the result can be used to decrease µ indefinitely, with For λ = n a positive integer, the contour in Eq. (72) can be closed, and the Cauchy residue theorem gives the expression as expected from Eq. (35). For ℜµ sufficiently negative, we can use the inverse operator M −λ − to increase µ. In particular, writing M −n − as an n-fold product M −1 − · · · M −1 − gives the multi-integral expression ℜ(−n − ν − µ) > 0. Weyl-type relations for P µ ν were not considered in [1]; the results in this section are new. The only natural endpoint for the Legendre functions z 2 − 1 ±µ/2 F µ ν (z) for arbitrary z is at z = 1. This leads to integrals of the form for M λ ± , with endpoints at t 0 = ∓(z − 1), with the condition that the behavior of the functions near the endpoints is such that potential endpoint contributions to the integral vanish [13].
with the initial endpoint at t 0 = e iπ (z − 1) on the first sheet of the cut t plane. The substitution t = e iπ t ′ converts the original contour to (z − 1, 0+, z − 1) with the cut in t ′ along the positive real axis. We can eliminate the dependence of the endpoint on z through the scaling t ′ = (z − 1)v. This gives The operator M λ + increases the order µ of the Legendre function on which it acts to µ + λ. To arrange that the leading factor of (z − 1) appear with the power −(µ + λ) instead of −µ, we can use the identity to extract a factor (z − 1) −µ from the integrand, and rearrange the result to get The integral converges with vanishing endpoint contributions for ℜµ < 1.
To show directly that the expression on the right-hand sides of Eq. (78) is an associated Legendre function of order µ+λ and degree ν, we can demonstrate that it satisfies the the corresponding differential equation. The input function The application of the operator in Eq. (79) with µ → µ + λ to the expression in Eq. (78) leads to the action of the operator in Eq. (80), again with µ → µ + λ, on the integral. We can eliminate the term in ν(ν + 1) by noting that the V -dependent factors in the integrand give w µ ν (V ) and then using Eq. (80) for that function. We find that the resulting integrand, expressed in terms of the integration variable v, can be written as derivative, giving where we have evaluated the integral at the endpoints v = e 2πi and v = 1. The expression in Eq. (82) vanishes for v → 1 for ℜµ < 1, so the right-hand side of Eq. (78) must be an associated Legendre function (z 2 − 1) −(µ+λ)/2 F µ+λ ν (z). The leading factor (z − 1) −µ−λ in Eq. (78) gives the proper behavior of (z 2 − 1) −(µ+λ)/2 P µ+λ ν (z) for z → 1. The limit in the case of (z 2 − 1) −(µ+λ)/2 Q µ+λ ν involves an additional constant term which is absent in Eq. (78). The function F µ+λ ν must therefore be proportional to P µ+λ ν with no admixture of Q µ+λ ν . We can determine the constant of proportionality, e −iπλ , by using the asymptotic behavior of (z 2 −1) −(µ+λ)/2 P µ+λ ν (z) and the integral for z → ∞, with the final result that ℜµ < 1. There are no restrictions on λ in this form; however, the contour integral can only be collapsed to a simple integral over the range 0 ≤ v ≤ 1 for ℜλ < 0.
With this result established, we can return to Eq. (76), make the substitution t → e iπ t as before, and, incorporating the final phase, rewrite that expression in the equivalent forms The integrals converge for ℜµ < 1. The second form can be identified as the analytic continuation in λ of a known Riemann fractional integral ( [6], Eq. 13.1(52), with γ = 2, u = 2y + 1); the latter is valid only for ℜλ < 0. As was the case for the Weyl-type representations of M λ + , the special cases λ = n and λ = −n, n a positive integer, lead to repeated derivative and integral expressions. For λ = n, we can close the contour in Eq. (84) and use the Cauchy residue theorem to obtain as expected from Eq. (33). For λ = −n, we can write M −n + as the n-fold product M −1 + · · · M −1 + and follow the procedure that led to Eq. (49) to obtain the multiple integral These results can be used to obtain simple relations for the functions P µ ν analogous to those for Q µ ν in Eqs. (50) to (53), with the difference that the factors (−1) n multiplying the integrals in the latter are absent in the former, the result of the shift from Weyl-to Riemann-type integrals.
In the case of the functions (z 2 − 1) −(µ+λ)/2 Q µ+λ ν (z), the expression in Eq. (82) with P µ ν (V ) replaced by Q µ ν (V ) has a finite limit for v → 1 with ℜµ < 1. As a result, the function defined by the Q µ ν version of Eq. (78) satisfies an inhomogeneous version of the Legendre equation rather than that equation itself. We will not consider this case in detail. However, the integral in Eq. (76) converges for ℜµ < 1 and can be evaluated using the expression for Q µ ν in Eq. (16) after an Euler transformation on the second term, giving the new result The calculation follows the procedures used in the following subsection for the case of M λ − z 2 − 1 µ/2 P µ ν (z).

Relations for M λ −
Riemann-type relations do not hold for the action of M λ − on the natural functions z 2 − 1 µ/2 F µ ν (z), Eq. (34). To show this, we start with Eq. (75), introduce the variable V = z − (z − 1)v, and factor out a power (z − 1) µ so that the leading factor involves the expected power (z − 1) µ−λ . This gives the expressions A calculation similar to that following Eq. (78) with µ replaced by −µ in Eq. (79), and f µ ν = z 2 − 1 µ/2 F µ ν , gives the condition for the right-hand side of Eq. (89) to satisfy the associated Legendre equation. This requires that the expression vanish. It does not vanish for either P µ ν or Q µ ν , reducing to for F µ ν = P µ ν , to the same result for Q µ ν with µ > −1, and diverging for Q µ ν for µ < −1.
is therefore a solution of the inhomogeneous Legendre equation rather than the homogeneous equation. In particular, the integral is not simply proportional to the function (z) as would be expected had the Riemann version of M λ − commuted with the Legendre operator M 2 + ν(ν + 1) as in Eq. (40).
Equations (89)-(91) give the general result for the Riemann form for M λ − for non-integer λ. We will evaluate the integral directly using the form in Eq. (90) and the expression for P µ ν in Eq. (14). This gives where we have used the relation (1 − V ) = (1 − v)(1 − z) in the second line and identified the remaining integrals in terms of the analytic continuation of the beta function B(−λ, k + 1), The series in Eq. (97) converges for |(z − 1)/2| < 1, the same region as the hypergeometric representation we used for P µ ν . The corresponding result for F µ ν = Q µ ν in Eq. (90) can be obtained by the same method, but involves extra terms from the second line in Eq. (16). We will not consider it here.
In the special case λ = n a positive integer, we can close the contour in Eq. (89) and use the residue theorem to obtain the differential relations for F µ ν either P µ ν or Q µ ν . The second line in these relations follows directly from Eq. (35). It can also be obtained for P µ ν from Eq. (98) by noting that the first n terms in the expansion in Eq. (97) vanish, shifting the summation index appropriately, and using the expression for P µ ν in Eq. (14). For the inverse relation with λ = −n, we can collapse the contour in Eq. (89), write M −n − as an n-fold product, . . , u n = u n−1 − t n to obtain the multi-integral representations valid for F µ ν = P µ ν for all µ, and for Q µ ν for µ > −1. The multi-integral in Eq. (102), which is likely to appear in some physical applications, was evaluated directly by Cohl and Costas-Santos [1] in the case of in agreement with the general expression in Eq. (98). The same structure would be expected for Q µ ν . The result for M λ − z 2 − 1 µ/2 P µ ν (z) in Eq. (98) holds for all λ. To investigate the behavior of this function for z large, we will introduce a Barnes-type representation for the 3 F 2 and rewrite Eq. (103) as where the contour C runs from −i∞ to i∞, passing to the left of s = 0 and to the right of s = −1 and the poles of the integrand at s = ν + µ and −ν + µ − 1, taking ℑµ < 0 if necessary to displace the poles from the real axis. The integral converges for |arg(z − 1)| < π.
Taking ℜ(z − 1) > 0, we can close the contour to the left and evaluate the integral in terms of the sums of the residues at the poles at s = ν + µ − n, −ν + µ − n − 1, −n, n = 0, . . . , ∞. After using the reflection formula for the gamma function to convert −n to n in the gamma functions in the residues, we obtain The functions in first two parts of this expression, (105a) and (105b), are multiples of 3.2(37) after an Euler transformation which eliminates powers of (z + 1) in the final result). These are, of course, solutions of the associated Legendre equation, and can be continued without difficulty to the region with |(z − 1)/2| < 1. The final term in (105c) is not a solution of the associated Legendre equation, but is rather a particular solution of the inhomogeneous equation in Eq. (94). We note in particular that the asymptotic form of that term for z → ∞ has the indicial behavior z −λ−1 expected from the result in Eq. (94), and not z ν+µ−λ or z −ν+µ−λ−1 as would be the case for an associated Legendre function The expressions in (105a) and (105b) can also be written in terms of Legendre functions of the first and second kind using [9] 3.2(19), again with Euler transformations in those expressions, with the result The ratio of gamma functions in the second term in Eq. (106) is the same as the ratio in the first term, with ν replaced by −ν − 1 in the latter. This reflects the transformation from the leading term The form of the particular solution of the inhomogeneous Legendre equation in Eq. (105c) in the region with |(z − 1)/2| < 1 follows by analytic continuation, and is simply the the expression in Eq. (98) with the first two terms in Eq. (106) subtracted from it. This was obtained in [1] in the special case λ = −n by n-fold integration as in Eq. (102). The term proportional to Q µ−λ ν vanishes in this case, and the result agrees with that of Cohl and Costas-Santos, where the particular solution arose from the non-zero endpoint contributions in the repeated integrations.

IV. CHANGE OF THE DEGREE OF
A. Weyl-and Riemann-type relations for K λ

3
We studied the action of the degree-raising operators K λ 3 on the functions (y 2 − 1) −(λ+1)/2 F µ ν (y/ y 2 − 1) defined in Eq. (38) in [3], Sec. VIII A, using the original form of the operator K 3 . We give a simpler treatment here using the derivative form of K 3 acting on those functions as in Eq. (39). This parallels the treatment of M λ − in Sec. III B as expected from the automorphism noted in [14].
We begin with the general Weyl-type relation for K λ 3 which follows from the action of K 3 as a simple derivative on the functions in Eq. (39), where the contour of integration must be picked appropriately to avoid the singularities in the integrand. We will follow the procedure used for M λ − in Sec. III B, assume that the contour initially runs above the branch points at t = z ± 1, rotate the contour clockwise by π, and redefine the integration variable as there to obtain with Y = y + t. The integral converges for ℜ(ν + λ − µ + 1) > 0 for F µ ν = P µ ν , and for ℜ(ν + λ ± µ + 1) > 0 for Q µ ν . At this point, we use a Whipple transformation ( [9], Eqs. 3. 3.1(13,14); [10], Eqs. 14.9 (16,17)) on the Legendre function on the right-hand side of Eq. (108) to write Choosing F µ ν = P µ ν in Eq. (108) and using Eq. (109), we obtain where the last line follows from Eqs. (65) and (66). Using Eq. (109) in the reverse sense then gives ℜ(ν + λ − µ + 1) > 0, in agreement with the corresponding result in [3], Eq. (100).
A similar calculation starting with the associated Legendre functions of the second kind gives the relations with convergence for ℜ(ν + λ ± µ + 1) > 0. In each case, K 3 acts as a raising operator on ν as expected. The intermediate steps as in Eq. (111) also illustrate the effect of the inner automorphism of the underlying algebra of e(2,1) discussed in [14], with the action of K 3 on the Legendre functions F µ ′ ν ′ y/ y 2 − 1 appearing as the action of M − on F µ ν (y), with ν ′ = −µ − 1 2 and µ ′ = −ν − 1 2 . For λ = n in Eqs. (111) and (114), the integrals can be evaluated using the residue theorem, giving the multiderivative relations in accord with Eq. (39). We can also obtain an interesting multi-integral representation for the Legendre functions by taking λ = −n, and writing K −n 3 as an n-fold product of K −1 3 . This gives There are no Riemann-type representations for K λ 3 which reproduce the input Legendre function F µ ν with the expected shifted degree ν + λ. The natural endpoint for a Riemann representation in terms of the original integrand in Eq. (107) is at t = y − 1. Writing t = (y − 1)u, that equation can be rewritten as with U = y − (y − 1)u. If we use the appropriate Whipple transformations from Eq. (109) or Eq. (110) to replace , we obtain a numerical multiple of the expression in Eq. (91) for the action of M λ − on F ′ µ ′ ν ′ on the same contour. As already shown in Sec. III C 2, the resulting integrands do not vanish sufficiently rapidly for u → 1 for the Riemann integrals to satisfy the homogeneous Legendre equation.
In the case of the input function Q µ ν , the integral involving the Whipple-transformed function P µ ′ ν ′ is of the form in Eq. (90) with ν, µ replaced by ν ′ = −µ − 1 2 , µ ′ = −ν − 1 2 . The resulting function is a solution of the inhomogeneous Legendre equation in Eq. (94), with the transformed indices, rather than the homogeneous Legendre equation. The integral can be evaluated as in Eqs. (95) and following, with the result We will not consider this further.
B. Weyl-and Riemann-type relations for P λ 3 The treatment of P λ 3 largely parallels that of K λ 3 . The automorphism of the underlying e(2,1) algebra in [14] connects P 3 to M + , so in calculations such as those above, the use of the Whipple transformation connects the action of P λ 3 on Legendre functions of argument y/ y 2 − 1 to that of M λ + on functions of y. We begin with the expression After a rotation of the contour of integration and redefinition of the integration variable as in the transition from Eq. (107) to Eq. (108), this becomes again with Y = y + t. The integrals converge for ℜ(ν + µ − λ) < 0 for F µ ν = P µ ν , and for ℜ(ν ± µ − λ) < 0 for F µ ν = Q µ ν . In the case of P µ ν , we can take ℜν > − 1 2 by the symmetry of those functions under the transformation ν → −ν − 1. Convergence then requires that − 1 2 < ℜν < ℜ(λ − µ), a constraint on ℜλ for given ν and µ. If we use the Whipple transformation on P µ ν in Eq. (109), we obtain a Weyl-type integral for the action of M λ Eq. (42), changing the order from −ν − 1 2 to −ν +λ− 1 2 . Using that result and the inverse of the Whipple transformation in Eq. (109), we find that a result obtained in [3], Eq. (110), through a less direct calculation. For F µ ν = Q µ ν in Eq. (121), the intermediate step in the calculation above involves M λ + acting on a Legendre function of the first kind. As shown above in Sec. III A 2, Eq. (57), the result involves a combination of functions of the first and second kinds. This structure is preserved by the inverse Whipple transformation used to get the final result. This does not appear to be of much practical importance, and will be omitted here.
For λ = n a positive integer such that n > ℜ(ν + µ), ℜν > − 1 2 , we can close the integration contour in Eq. (122) and evaluate the integral using the residue theorem. This gives d dy a result which also follows directly from Eq. (37). The integrals in the inverse relation with P λ 3 replaced by P −λ 3 , ℜλ > 0, converge only for ℜ(ν +µ+λ) < 0, ℜν > − 1 2 , and require large negative values of ℜµ to obtain a comparable range in ℜλ. This limits the range in n in which P −n 3 can be written as a multi-integral of the type encountered earlier, and we will skip that result.
Returning to the integrand in Eq. (120), it is evident that the natural endpoint for a Riemann-rather than Weyltype expression for P λ 3 is at t = y − 1 where the argument of the Legendre function diverges. If we let t = (y − 1)u, that integral becomes with U = y − (y − 1)u as before. The integrand must vanish sufficiently rapidly at the endpoints of the contour that there are no endpoint contributions. This is not the case for F µ ν = P µ ν as previously established in [3], Sec. VIII B, so there is no proper Riemann-type representation for P λ 3 . The function defined by the integral is actually a solution of an inhomogeneous version of the Legendre equation for the function y 2 − 1 (ν−λ)/2 P µ ν−λ y/ y 2 − 1 , and can presumably be treated after a Whipple transformation using the methods of Sec. III C 1. We will not pursue this.
With the choice F µ ν = Q µ ν in Eq. (125), we obtain the Riemann-type relation The integral converges with vanishing endpoint contributions for ℜν > − 3 2 . A Whipple transformation using Eq. (110) leads to an expression equivalent to that for the action of M λ + on the functions y 2 − 1 −µ ′ /2 P µ ′ ν ′ (y) with ν ′ = −µ − 1 2 and µ ′ = −ν − 1 2 treated in Sec. III C 1; see especially Eq. (84). Using the inverse Whipple transformation on this result, we find that For λ = n, the contour can be closed and the residue theorem gives d dy as expected from Eq. (37). The inverse relation with λ = −n gives These results are new.

V. SO(3) AND THE FERRERS FUNCTIONS
The Ferrers functions, the usual associated Legendre functions "on the cut" with −1 < x < 1, are defined in terms of the functions in the complex plane as ( [9], Eqs. 3.4(1, 2); [10], Eqs. 14.23 (4,5) ) These functions appear naturally in the treatment of the usual angular momentum algebra of SO(2) described by operators L 1 , L 2 , L 3 with the commutation relations [L i , L j ] = iǫ ijk L k ; see, for example, [4,14,15]. The operator L 2 = L 2 1 + L 2 2 + L 2 3 gives the associated Legendre equation in x = cos θ, while L ± = L 1 ± iL 2 act as raising and lowering operators on the order µ of the P µ ν (x) and Q µ ν (x). The full operators in terms of the variables t = e iφ and x = cos θ are acting on functions t µ F µ ν (x), F µ ν a Ferrers function. After factoring out the t dependence, these become the differential recurrence relations for the latter, In this form, L + = −iM + (x + i0) and L − = −iM − (x + i0). These operators reduce to simple derivatives when acting on modified functions f µ ν (x), with Because the interval −1 < x < 1 is bounded, the natural fractional operators L λ ± are of the Riemann type. As seen in Sec. III C, a Riemann-type relation exists for M λ + z 2 − 1 −µ/2 P µ ν (z), but not for M λ + z 2 − 1 −µ/2 Q µ ν (z), the latter leading to a solution of an inhomogeneous rather than homogeneous Legendre equation. The result is similar for L λ + , with L λ + 1 − x 2 −µ/2 P µ ν (x) well-defined, but again no proper Riemann-type relation in the case of Q µ ν (x). In particular, using the results in Eqs. (139) and (140), we find that L λ + 1 − x 2 −µ/2 P µ ν (x) = 1 2πi e iπλ Γ(λ + 1) (1−x,0+,1−x) dt t λ+1 1 − (x + t) 2 −µ/2 P µ ν (x + t) ℜµ < 1. This result gives the analytic continuation in λ of a known Riemann fractional integral ( [6], Eq. 13.1(54)) obtained by collapsing the contour in Eq. (143) for ℜλ < 0 to an integration from 0 to 1−x and shifting the integration variable to v = x + t. That result is only valid for ℜλ < 0. By closing the contour for λ = n a positive integer, or writing L −n + as an n-fold product of L −1 + , we immediately obtain the derivative and multi-integral representations discussed in [1], Eqs. (64) results useful in angular momentum related problems. The result for L λ + 1 − x 2 −µ/2 Q µ ν (x) is similar to that for M λ + z 2 − 1 −µ/2 Q µ ν (z) in Eq. (88), with This result follows directly from Eq. (147) by using the expression for Q µ ν in Eq. (134) after an Euler transformation on the first term, followed by the substitution t = (1 − x)v, and term-by-term integration. Cohl and Costas-Santos ( [1], Eq. (76)) derived the special case of this relation with λ = −n and µ → −µ by repeated integrations taking the non-vanishing endpoint terms in Eq. (147) into account.