On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)
aa r X i v : . [ m a t h . C A ] O c t On parameter derivatives of the associated Legendre functionof the first kind (with applications to the construction of theassociated Legendre function of the second kind of integerdegree and order)
Rados law SzmytkowskiAtomic Physics Division, Department of Atomic Physics and Luminescence,Faculty of Applied Physics and Mathematics, Gda´nsk University of Technology,Narutowicza 11/12, PL 80–233 Gda´nsk, Polandemail: [email protected] 20, 2018
Abstract
A relationship between partial derivatives of the associated Legendre function of the firstkind with respect to its degree, [ ∂P mν ( z ) /∂ν ] ν = n , and to its order, [ ∂P µn ( z ) /∂µ ] µ = m , is es-tablished for m, n ∈ N . This relationship is used to deduce four new closed-form representa-tions of [ ∂P mν ( z ) /∂ν ] ν = n from those found recently for [ ∂P µn ( z ) /∂µ ] µ = m by the present author[R. Szmytkowski, J. Math. Chem. 46 (2009) 231]. Several new expressions for the associatedLegendre function of the second kind of integer degree and order, Q mn ( z ), suitable for numericalpurposes, are also derived. KEY WORDS:
Legendre functions; parameter derivatives; special functions
MSC2010:
Recently, an interest has arisen in the derivation of closed-form expressions for parameter deriva-tives of the associated Legendre functions [1–4]. In Refs. [3, 4], the present author has extensivelystudied such derivatives for the associated Legendre function function of the first kind, P µν ( z ), in thecase when one of the parameters is a fixed integer. In particular, in Ref. [3], using finite-sum expres-sions for P µn ( z ), with n ∈ N , we have found the following two representations of [ ∂P µn ( z ) /∂µ ] µ = m ,with m ∈ N : ∂P µn ( z ) ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ = m = 12 P mn ( z ) ln z + 1 z −
1+ ( − ) m (cid:18) z + 1 z − (cid:19) m/ m − X k =0 ( − ) k ( k + n )!( m − k − k !( n − k )! (cid:18) z − (cid:19) k + (cid:18) z − (cid:19) m/ n − m X k =0 ( k + n + m )! ψ ( k + 1) k !( k + m )!( n − m − k )! (cid:18) z − (cid:19) k (0 m n )(1.1)1nd ∂P µn ( z ) ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ = m = 12 P mn ( z ) ln z + 1 z − ψ ( n + m + 1) + ψ ( n − m + 1)] P mn ( z ) − ( − ) n ( n + m )!( n − m )! (cid:18) z + 1 z − (cid:19) m/ n X k =0 ( − ) k ( k + n )! ψ ( k + m + 1) k !( k + m )!( n − k )! (cid:18) z + 12 (cid:19) k (0 m n ) . (1.2)In turn, in Ref. [4], using contour-integral representations of ∂P mν ( z ) /∂ν , we have arrived, amongothers, at the following three expressions for [ ∂P mν ( z ) /∂ν ] ν = n , again with m, n ∈ N : ∂P mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = P mn ( z ) ln z + 12 − [ ψ ( n + 1) + ψ ( n − m + 1)] P mn ( z )+ (cid:18) z − (cid:19) m/ n − m X k =0 ( k + n + m )! ψ ( k + n + m + 1) k !( k + m )!( n − m − k )! (cid:18) z − (cid:19) k + ( n + m )!( n − m )! (cid:18) z − z + 1 (cid:19) m/ n X k =0 ( k + n )! ψ ( k + n + 1) k !( k + m )!( n − k )! (cid:18) z − (cid:19) k (0 m n ) , (1.3) ∂P mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = P mn ( z ) ln z + 12 + [ ψ ( n + 1) − ψ ( n − m + 1)] P mn ( z ) − ( − ) n ( n + m )!( n − m )! (cid:18) z − (cid:19) − m/ m − X k =0 ( k + n − m )!( m − k − k !( n + m − k )! (cid:18) z + 12 (cid:19) k + ( − ) n + m (cid:18) z − (cid:19) m/ n − m X k =0 ( − ) k ( k + n + m )! k !( k + m )!( n − m − k )! × [ ψ ( k + n + m + 1) − ψ ( k + m + 1)] (cid:18) z + 12 (cid:19) k + ( − ) n ( n + m )!( n − m )! (cid:18) z + 1 z − (cid:19) m/ n X k =0 ( − ) k ( k + n )! k !( k + m )!( n − k )! × [ ψ ( k + n + 1) − ψ ( k + 1)] (cid:18) z + 12 (cid:19) k (0 m n ) (1.4)and ∂P mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = P mn ( z ) ln z + 12 + [ ψ ( n + m + 1) − ψ ( n + 1)] P mn ( z ) − ( − ) n + m (cid:18) z − z + 1 (cid:19) m/ m − X k =0 ( k + n )!( m − k − k !( n − k )! (cid:18) z + 12 (cid:19) k + ( − ) n + m (cid:18) z − (cid:19) m/ n − m X k =0 ( − ) k ( k + n + m )! k !( k + m )!( n − m − k )! × [ ψ ( k + n + m + 1) − ψ ( k + 1)] (cid:18) z + 12 (cid:19) k + ( − ) n ( n + m )!( n − m )! (cid:18) z + 1 z − (cid:19) m/ n X k =0 ( − ) k ( k + n )! k !( k + m )!( n − k )! × [ ψ ( k + n + 1) − ψ ( k + m + 1)] (cid:18) z + 12 (cid:19) k (0 m n ) . (1.5)2n the above equations, and in what follows, ψ ( ζ ) is the digamma function defined as ψ ( ζ ) = 1Γ( ζ ) dΓ( ζ )d ζ . (1.6)In the present paper, we shall pursue further the subject of derivation of closed-form expressionsfor parameter derivatives of the associated Legendre function of the first kind. First, in Section 2,we shall show that there exists a simple relationship between the derivatives [ ∂P mν ( z ) /∂ν ] ν = n and[ ∂P µn ( z ) /∂µ ] µ = m , both with m, n ∈ N . Next, in Section 3, we shall use this relationship, in conjunc-tion with the formulas (1.1) and (1.2), to derive four further representations of [ ∂P mν ( z ) /∂ν ] ν = n ,two of them involving sums of powers of ( z + 1) / z − /
2. Interestingly, each out of these four representations contains only two sums. Therefore,the two new expressions for [ ∂P mν ( z ) /∂ν ] ν = n containing sums of powers of ( z + 1) / z − / ψ ( n + m + 1) − ψ ( n + 1)] P mn ( z ). In the final Section 4, we shall exploitthe results of Section 3 to find some new representations of the associated Legendre function of thesecond kind of integer degree and order, Q mn ( z ), suitable for use for numerical purposes in variousparts of the complex z -plane.Throughout the paper, we shall be adopting the standard convention according to which z ∈ C \ [ − , − π < arg( z ) < π, − π < arg( z ± < π (1.7)(this corresponds to drawing a cut in the z -plane along the real axis from z = −∞ to z = +1),hence, − z = e ∓ i π z − z + 1 = e ∓ i π ( z − − z − ∓ i π ( z + 1) (arg( z ) ≷ . (1.8)Also, it will be implicit that x ∈ [ − , µ, ν ∈ C and k, m, n ∈ N . Finally, it will be understoodthat if the upper limit of a sum is less by unity than the lower one, then the sum vanishes identically.The associated Legendre functions of the first and the second kinds used in the paper are those ofHobson [5] (cf. also Refs. [6–8]).Before proceeding to the matter, we emphasize that the results obtained in the present pa-per are interesting not only for their own (mathematical) sake. The derivatives ∂P mν ( z ) /∂ν and ∂P mν ( x ) /∂ν are met in solutions of some boundary value problems of theoretical acoustics, electro-magnetism, heat conduction and other branches of theoretical physics and applied mathematics (fora list of references illustrating this statement, see Ref. [4]). Physical applications of the associatedLegendre functions of the second kind of integer degree and order, Q mn ( z ) and Q mn ( x ), (for which,recall, several new explicit expressions are derived in Section 4) are even more abundant [9, 10]. [ ∂P mν ( z ) /∂ν ] ν = n and [ ∂P µn ( z ) /∂µ ] µ = m The departure point for our considerations in this section is the following Rodrigues-type formula,due to Barnes [11], for the associated Legendre function of the first kind when the sum of its degreeand its order is a non-negative integer: P m − νν ( z ) = 12 ν Γ( ν + 1) ( z − ( m − ν ) / d m ( z − ν d z m . (2.1)In terms of the Jacobi polynomial P ( α,β ) n ( z ) = 12 n n ! ( z − − α ( z + 1) − β d n d z n (cid:2) ( z − n + α ( z + 1) n + β (cid:3) ( α, β ∈ C ) , (2.2)3q. (2.1) may be rewritten as P m − νν ( z ) = m !Γ( ν + 1) (cid:18) z − (cid:19) ( ν − m ) / P ( ν − m,ν − m ) m ( z ) . (2.3)(With no doubt, the reader has immediately realized that the Jacobi polynomial appearing on theright-hand side of Eq. (2.3) is a multiple of the Gegenbauer polynomial C ( ν − m +1 / m ( z ). However,we shall not make any use of this fact here.) Differentiation of Eq. (2.3) with respect to ν , followedby setting ν = n , yields ∂P m − nν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n − ∂P µn ( z ) ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ = m − n = 12 P m − nn ( z ) ln z − − ψ ( n + 1) P m − nn ( z )+ m ! n ! (cid:18) z − (cid:19) ( n − m ) / ∂P ( λ,λ ) m ( z ) ∂λ (cid:12)(cid:12)(cid:12)(cid:12) λ = n − m . (2.4)The replacement of m by n + m results in the relationship ∂P mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n − ∂P µn ( z ) ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ = m = 12 P mn ( z ) ln z − − ψ ( n + 1) P mn ( z )+ ( n + m )! n ! (cid:18) z − (cid:19) − m/ ∂P ( λ,λ ) n + m ( z ) ∂λ (cid:12)(cid:12)(cid:12)(cid:12) λ = − m . (2.5)If in Eq. (2.5) one exploits the following two explicit representations of the Jacobi polynomial P ( α,β ) n ( z ): P ( α,β ) n ( z ) = Γ( n + α + 1)Γ( n + α + β + 1) n X k =0 Γ( k + n + α + β + 1) k !( n − k )!Γ( k + α + 1) (cid:18) z − (cid:19) k , (2.6) P ( α,β ) n ( z ) = ( − ) n Γ( n + β + 1)Γ( n + α + β + 1) n X k =0 ( − ) k Γ( k + n + α + β + 1) k !( n − k )!Γ( k + β + 1) (cid:18) z + 12 (cid:19) k , (2.7)after making use of Eq. (2.3) and of the easily provable relationlim λ →− m ψ ( k + λ + 1)Γ( k + λ + 1) = ( − ) k + m ( m − k − k m − , (2.8)one obtains ∂P mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n − ∂P µn ( z ) ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ = m = 12 P mn ( z ) ln z − − ψ ( n − m + 1) P mn ( z ) − ( − ) m ( n + m )!( n − m )! (cid:18) z − (cid:19) − m/ m − X k =0 ( − ) k ( k + n − m )!( m − k − k !( n + m − k )! (cid:18) z − (cid:19) k + ( n + m )!( n − m )! (cid:18) z − z + 1 (cid:19) m/ n X k =0 ( k + n )! k !( k + m )!( n − k )! × [2 ψ ( k + n + 1) − ψ ( k + 1)] (cid:18) z − (cid:19) k (0 m n ) (2.9)4nd ∂P mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n − ∂P µn ( z ) ∂µ (cid:12)(cid:12)(cid:12)(cid:12) µ = m = 12 P mn ( z ) ln z − − ψ ( n − m + 1) P mn ( z ) − ( − ) n ( n + m )!( n − m )! (cid:18) z − (cid:19) − m/ m − X k =0 ( k + n − m )!( m − k − k !( n + m − k )! (cid:18) z + 12 (cid:19) k + ( − ) n ( n + m )!( n − m )! (cid:18) z + 1 z − (cid:19) m/ n X k =0 ( − ) k ( k + n )! k !( k + m )!( n − k )! × [2 ψ ( k + n + 1) − ψ ( k + 1)] (cid:18) z + 12 (cid:19) k (0 m n ) , (2.10)respectively. [ ∂P mν ( z ) /∂ν ] ν = n with m n In this section, we shall use the results of Section 2 to provide several expressions for the derivative[ ∂P mν ( z ) /∂ν ] ν = n , which differ from these given in Eqs. (1.3)–(1.5).The first from among these expressions for [ ∂P mν ( z ) /∂ν ] ν = n follows if one plugs the represen-tation (1.2) of [ ∂P µn ( z ) /∂µ ] µ = m into the relationship (2.10). The result ∂P mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = P mn ( z ) ln z + 12 + [ ψ ( n + m + 1) − ψ ( n − m + 1)] P mn ( z ) − ( − ) n ( n + m )!( n − m )! (cid:18) z − (cid:19) − m/ m − X k =0 ( k + n − m )!( m − k − k !( n + m − k )! (cid:18) z + 12 (cid:19) k + ( − ) n ( n + m )!( n − m )! (cid:18) z + 1 z − (cid:19) m/ n X k =0 ( − ) k ( k + n )! k !( k + m )!( n − k )! × [2 ψ ( k + n + 1) − ψ ( k + m + 1) − ψ ( k + 1)] (cid:18) z + 12 (cid:19) k (0 m n )(3.1)is seen to be much simpler than either of the representations (1.4) or (1.5). An infinite variety ofother representations of [ ∂P mν ( z ) /∂ν ] ν = n , involving sums of powers of ( z + 1) /
2, may be obtainedby taking linear combinations, with coefficients such that their sum is unity, of the expressions inEqs. (1.4), (1.5) and (3.1). For instance, multiplying Eq. (3.1) by − ∂P mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = P mn ( z ) ln z + 12 − ( − ) n + m (cid:18) z − z + 1 (cid:19) m/ m − X k =0 ( k + n )!( m − k − k !( n − k )! (cid:18) z + 12 (cid:19) k + ( − ) n + m (cid:18) z − (cid:19) m/ n − m X k =0 ( − ) k ( k + n + m )! k !( k + m )!( n − m − k )! × [2 ψ ( k + n + m + 1) − ψ ( k + m + 1) − ψ ( k + 1)] (cid:18) z + 12 (cid:19) k (0 m n ) . (3.2)5t is seen that for m = 0 both the representations (3.1) and (3.2) of [ ∂P mν ( z ) /∂ν ] ν = n reduce to theformula ∂P ν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = P n ( z ) ln z + 12 + 2 n X k =0 ( − ) k + n ( k + n )!( k !) ( n − k )! [ ψ ( k + n + 1) − ψ ( k + 1)] (cid:18) z + 12 (cid:19) k , (3.3)found by the present author in Ref. [12, Section 5.2.7] (cf. also Ref. [13]).From the above findings, one may deduce two interesting and, as we shall see in a moment,useful identities involving the function P mn ( z ). If we equate the right-hand sides of Eqs. (1.4) and(3.1), this results in the first of these relations:[ ψ ( n + m + 1) − ψ ( n + 1)] P mn ( z ) = ( − ) n + m (cid:18) z − (cid:19) m/ n − m X k =0 ( − ) k ( k + n + m )! k !( k + m )!( n − m − k )! × [ ψ ( k + n + m + 1) − ψ ( k + m + 1)] (cid:18) z + 12 (cid:19) k − ( − ) n ( n + m )!( n − m )! (cid:18) z + 1 z − (cid:19) m/ n X k =0 ( − ) k ( k + n )! k !( k + m )!( n − k )! × [ ψ ( k + n + 1) − ψ ( k + m + 1)] (cid:18) z + 12 (cid:19) k (0 m n ) . (3.4)Replacement of z by − z in the above equation, followed by the use of the well-known property P mn ( − z ) = ( − ) n P mn ( z ) (0 m n ) (3.5)and also of Eq. (1.8), leads to the second identity:[ ψ ( n + m + 1) − ψ ( n + 1)] P mn ( z ) = (cid:18) z − (cid:19) m/ n − m X k =0 ( k + n + m )! k !( k + m )!( n − m − k )! × [ ψ ( k + n + m + 1) − ψ ( k + m + 1)] (cid:18) z − (cid:19) k − ( n + m )!( n − m )! (cid:18) z − z + 1 (cid:19) m/ n X k =0 ( k + n )! k !( k + m )!( n − k )! × [ ψ ( k + n + 1) − ψ ( k + m + 1)] (cid:18) z − (cid:19) k (0 m n ) . (3.6)Playing with Eq. (1.3) and with the identity (3.6), one may obtain an infinite variety of rep-resentations of [ ∂P mν ( z ) /∂ν ] ν = n containing sums of powers of ( z − /
2. Two examples of suchrepresentations are ∂P mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = P mn ( z ) ln z + 12 − [ ψ ( n + m + 1) + ψ ( n − m + 1)] P mn ( z )+ (cid:18) z − (cid:19) m/ n − m X k =0 ( k + n + m )! k !( k + m )!( n − m − k )! × [2 ψ ( k + n + m + 1) − ψ ( k + m + 1)] (cid:18) z − (cid:19) k + ( n + m )!( n − m )! (cid:18) z − z + 1 (cid:19) m/ n X k =0 ( k + n )! ψ ( k + m + 1) k !( k + m )!( n − k )! (cid:18) z − (cid:19) k (0 m n ) (3.7)6nd ∂P mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = P mn ( z ) ln z + 12 + [ ψ ( n + m + 1) − ψ ( n + 1) − ψ ( n − m + 1)] P mn ( z )+ (cid:18) z − (cid:19) m/ n − m X k =0 ( k + n + m )! ψ ( k + m + 1) k !( k + m )!( n − m − k )! (cid:18) z − (cid:19) k + ( n + m )!( n − m )! (cid:18) z − z + 1 (cid:19) m/ n X k =0 ( k + n )! k !( k + m )!( n − k )! × [2 ψ ( k + n + 1) − ψ ( k + m + 1)] (cid:18) z − (cid:19) k (0 m n ) . (3.8)For m = 0, both Eqs. (3.7) and (3.8) reduce to the Schelkunoff’s formula [14] (cf. also Ref. [12,Section 5.2.6]) ∂P ν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = P n ( z ) ln z + 12 − ψ ( n + 1) P n ( z ) + 2 n X k =0 ( k + n )! ψ ( k + n + 1)( k !) ( n − k )! (cid:18) z − (cid:19) k . (3.9)From the representations of [ ∂P mν ( z ) /∂ν ] ν = n found above, one may construct counterpart rep-resentations for [ ∂P − mν ( z ) /∂ν ] ν = n , using the relationship [4] ∂P − mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = ( n − m )!( n + m )! ∂P mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n − [ ψ ( n + m + 1) − ψ ( n − m + 1)] P − mn ( z )(0 m n ) (3.10)and the well-known property P mn ( z ) = ( n + m )!( n − m )! P − mn ( z ) (0 m n ) . (3.11)Moreover, it does not offer any difficulty to derive counterpart expressions on the cut x ∈ [ − , ∂P ± mν ( x ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = e ± i πm/ ∂P ± mν ( x + i0) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = e ∓ i πm/ ∂P ± mν ( x − i0) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n = 12 (cid:20) e ± i πm/ ∂P ± mν ( x + i0) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n + e ∓ i πm/ ∂P ± mν ( x − i0) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n (cid:21) (3.12)together with x + 1 ± i0 = 1 + x, x − ± i0 = e ± i π (1 − x ) . (3.13)Concluding this section we note that, in principle, the relations (2.9) and (2.10) might be alsoused in the opposite direction, i.e., to construct representations for [ ∂P µn ( z ) /∂µ ] µ = m from thoseknown for [ ∂P mν ( z ) /∂ν ] ν = n . As it appears, however, that all expressions for [ ∂P µn ( z ) /∂µ ] µ = m obtainable in this way are much more complex (and thus potentially less useful) than these in Eqs.(1.1) and (1.2), we do not present them here. Q mn ( z ) with m n The associated Legendre function of the second kind, Q µν ( z ), may be defined [5] as the followinglinear combination of the Legendre functions of the first kind P µν ( z ) and P µν ( − z ): Q µν ( z ) = π i πµ e ∓ i πν P µν ( z ) − P µν ( − z )sin[ π ( ν + µ )] (Im( z ) ≷ . (4.1)7n the special case of µ = m , Eq. (4.1) simplifies to Q mν ( z ) = π ∓ i πν P mν ( z ) − P mν ( − z )sin( πν ) (Im( z ) ≷ , (4.2)hence, after exploiting the l’Hospital rule, one obtains Q mn ( z ) = ∓
12 i πP mn ( z ) + 12 ∂P mν ( z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n − ( − ) n ∂P mν ( − z ) ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ν = n (0 m n , Im( z ) ≷ . (4.3)Thus, we see that the problem of evaluation of Q mn ( z ) with 0 m n may be reduced to that ofderivation of expressions for [ ∂P mν ( ± z ) /∂ν ] ν = n .Accordingly, after combining Eq. (4.3) with Eqs. (1.3) and (3.1), we obtain Q mn ( z ) = 12 P mn ( z ) ln z + 1 z − ∓
12 [ ψ ( n + m + 1) + ψ ( n + 1)] P mn ( z ) ± ( ± ) n ( ∓ ) m n + m )!( n − m )! (cid:18) z − (cid:19) − m/ m − X k =0 ( ∓ ) k ( k + n − m )!( m − k − k !( n + m − k )! (cid:18) z ∓ (cid:19) k ± ( ± ) n + m (cid:18) z − (cid:19) m/ n − m X k =0 ( ± ) k ( k + n + m )! ψ ( k + n + m + 1) k !( k + m )!( n − m − k )! (cid:18) z ∓ (cid:19) k ∓ ( ± ) n n + m )!( n − m )! (cid:18) z ∓ z ± (cid:19) m/ n X k =0 ( ± ) k ( k + n )! k !( k + m )!( n − k )! × [ ψ ( k + n + 1) − ψ ( k + m + 1) − ψ ( k + 1)] (cid:18) z ∓ (cid:19) k (0 m n ) , (4.4)where the upper signs follow if [ ∂P mν ( z ) /∂ν ] ν = n is evaluated from Eq. (1.3) and [ ∂P mν ( − z ) /∂ν ] ν = n from Eq. (3.1), while the lower signs result if the roles of Eqs. (1.3) and (3.1) are interchanged.The same expression for Q mn ( z ) as above is obtained if Eq. (4.3) is coupled with Eqs. (1.4) and(3.7). Further, using Eqs. (1.3) and (3.2) in Eq. (4.3) leads to Q mn ( z ) = 12 P mn ( z ) ln z + 1 z − ∓
12 [ ψ ( n + 1) + ψ ( n − m + 1)] P mn ( z ) ± ( ± ) n ( − ) m (cid:18) z ± z ∓ (cid:19) m/ m − X k =0 ( ∓ ) k ( k + n )!( m − k − k !( n − k )! (cid:18) z ∓ (cid:19) k ∓ ( ± ) n + m (cid:18) z − (cid:19) m/ n − m X k =0 ( ± ) k ( k + n + m )! k !( k + m )!( n − m − k )! × [ ψ ( k + n + m + 1) − ψ ( k + m + 1) − ψ ( k + 1)] (cid:18) z ∓ (cid:19) k ± ( ± ) n n + m )!( n − m )! (cid:18) z ∓ z ± (cid:19) m/ n X k =0 ( ± ) k ( k + n )! ψ ( k + n + 1) k !( k + m )!( n − k )! (cid:18) z ∓ (cid:19) k (0 m n ) . (4.5)Next, if [ ∂P mν ( z ) /∂ν ] ν = n is obtained from Eq. (3.1) and [ ∂P mν ( − z ) /∂ν ] ν = n from Eq. (3.7), or vice8ersa, then Eq. (4.3) yields the expressions Q mn ( z ) = 12 P mn ( z ) ln z + 1 z − ∓ ψ ( n + m + 1) P mn ( z ) ± ( ± ) n ( ∓ ) m n + m )!( n − m )! (cid:18) z − (cid:19) − m/ m − X k =0 ( ∓ ) k ( k + n − m )!( m − k − k !( n + m − k )! (cid:18) z ∓ (cid:19) k ± ( ± ) n + m (cid:18) z − (cid:19) m/ n − m X k =0 ( ± ) k ( k + n + m )! k !( k + m )!( n − m − k )! × [2 ψ ( k + n + m + 1) − ψ ( k + m + 1)] (cid:18) z ∓ (cid:19) k ∓ ( ± ) n n + m )!( n − m )! (cid:18) z ∓ z ± (cid:19) m/ n X k =0 ( ± ) k ( k + n )! k !( k + m )!( n − k )! × [2 ψ ( k + n + 1) − ψ ( k + m + 1) − ψ ( k + 1)] (cid:18) z ∓ (cid:19) k (0 m n ) . (4.6)We are not aware of any appearance of either of the formulas (4.4)–(4.6) in the literature. Fur-thermore, if Eqs. (3.1) and (3.8) are used in Eq. (4.3), this results in Q mn ( z ) = 12 P mn ( z ) ln z + 1 z − ∓ ψ ( n + 1) P mn ( z ) ± ( ± ) n ( ∓ ) m n + m )!( n − m )! (cid:18) z − (cid:19) − m/ m − X k =0 ( ∓ ) k ( k + n − m )!( m − k − k !( n + m − k )! (cid:18) z ∓ (cid:19) k ± ( ± ) n + m (cid:18) z − (cid:19) m/ n − m X k =0 ( ± ) k ( k + n + m )! ψ ( k + m + 1) k !( k + m )!( n − m − k )! (cid:18) z ∓ (cid:19) k ± ( ± ) n n + m )!( n − m )! (cid:18) z ∓ z ± (cid:19) m/ n X k =0 ( ± ) k ( k + n )! ψ ( k + 1) k !( k + m )!( n − k )! (cid:18) z ∓ (cid:19) k (0 m n ) , (4.7)which is the same what follows if Eqs. (1.3) and (1.4) are plugged into Eq. (4.3) (cf. Ref. [4]).Finally, insertion of Eqs. (3.2) and (3.7) into Eq. (4.3) leads to Q mn ( z ) = 12 P mn ( z ) ln z + 1 z − ∓
12 [ ψ ( n + m + 1) + ψ ( n − m + 1)] P mn ( z ) ± ( ± ) n ( − ) m (cid:18) z ± z ∓ (cid:19) m/ m − X k =0 ( ∓ ) k ( k + n )!( m − k − k !( n − k )! (cid:18) z ∓ (cid:19) k ± ( ± ) n + m (cid:18) z − (cid:19) m/ n − m X k =0 ( ± ) k ( k + n + m )! ψ ( k + 1) k !( k + m )!( n − m − k )! (cid:18) z ∓ (cid:19) k ± ( ± ) n n + m )!( n − m )! (cid:18) z ∓ z ± (cid:19) m/ n X k =0 ( ± ) k ( k + n )! ψ ( k + m + 1) k !( k + m )!( n − k )! (cid:18) z ∓ (cid:19) k (0 m n ) , (4.8)which, in turn, is the same what is obtained if Eqs. (1.3) and (1.5) are coupled with Eq. (4.3) (cf.again Ref. [4]; for alternative derivations of the above result see Ref. [7, pages 81, 82 and 85] andRef. [3]). Other expressions for Q mn ( z ) may be obtained by combining Eqs. (4.4)–(4.8), with thepossible help of the identities (3.4) and (3.6).Once the function Q mn ( z ) is known, one may find the function Q − mn ( z ) from the well-knownrelationship Q − mn ( z ) = ( n − m )!( n + m )! Q mn ( z ) (0 m n ) . (4.9)9sing the formula Q ± mn ( x ) = ( − ) m h e ∓ i m/ Q ± mn ( x + i0) + e ± i m/ Q ± mn ( x − i0) i , (4.10)which follows from the Hobson’s [5] definition of the associated Legendre function of the secondkind on the cut x ∈ [ − , Q ± mn ( x ) with 0 m n . Since the procedure does not offer anydifficulty, we do not present the resulting expressions here. References [1] Yu. A. Brychkov, On the derivatives of the Legendre functions P µν ( z ) and Q µν with respect to µ and ν , Integral Transforms Spec. Funct. doi:10.1080/10652460903069660[2] H. S. Cohl, Derivatives with respect to the degree and order of associated Legendre functionsfor | z | > k z α / d z k = [Γ( α + 1) / Γ( α − k + 1)] z α − k with respect to αα