IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 40 (007) 14887 14891 doi:10.1088/1751-8113/40/49/00 ADDENDUM Addendum to On the derivative of the Legendre function of the first ind with respect to its degree Radosław Szmytowsi Atomic Physics Division, Department of Atomic Physics and Luminescence, Faculty of Applied Physics and Mathematics, Gdańs University of Technology, Narutowicza 11/1, PL 80 95 Gdańs, Poland E-mail: rade@mif.pg.gda.pl Received 16 July 007, in final form 31 August 007 Published 1 November 007 Online at stacs.iop.org/jphysa/40/14887 Abstract In our recent paper (Szmytowsi 006 J. Phys. A: Math. Gen. 39 15147; 007 J. Phys. A: Math. Theor. 40 7819 (corrigendum)), we have investigated the derivative of the Legendre function of the first ind, P ν (z), with respect to its degree. In this addendum, we derive some further representations of [ P ν (z)/ ν] ν=n with n N. The results are used to obtain in an elementary way two formulae for the Legendre function of the second ind of integer degree, Q n (z). PACS numbers: 0.30.Gp, 0.30.Lt In a recent paper [1], we have extensively investigated the derivative of the Legendre function of the first ind, P ν (z), with respect to its degree ν, with emphasis on the particular case of ν = n N. We have shown that [ P ν (z)/ ν] ν=n is of the form P ν (z) ν = P n (z) ln z 1 R n (z) (n N), (1) ν=n where P n (z) is the Legendre polynomial and R n (z) is a polynomial in z of degree n. Some previously nown formulae for R n (z), due to Bromwich and Schelunoff, have been rederived by us and a new expression has also been given. After the paper [1] has been published, it has come to our attention that we have not fully exploited the Jolliffe s formula P ν (z) ν = P n (z) ln z 1 1 d n ν=n n 1 n! dz n [ (z 1) n ln z 1 ] discussed in section 5.1 of [1]. In this addendum, it will be shown how this formula may be used to derive two further new representations of R n (z) and to rederive, in a much simpler way, the representation (5.90). We shall also obtain two expressions for the Legendre function of the second ind, Q n (z), different from those given in [1]. 1751-8113/07/491488705$30.00 007 IOP Publishing Ltd Printed in the UK 14887 ()
14888 Addendum We begin with the observation that from (1) and () it follows that d n [ (z 1) n (z 1) n ln z 1 dz n R n (z) = P n (z) ln z 1 1 n 1 n! Obviously, this may be rewritten as R n (z) = P n (z) ln z 1 1 n 1 n! ]. (3) n d n (z 1) n d [ (z 1) n ln z 1 ]. (4) dz n dz Preparing the next step, we provide here the following differentiation formula: d Ɣ(α 1) Ɣ(α 1) dz [zα ln z] = Ɣ(α 1) zα ln z [ψ(α 1) ψ(α 1)]zα Ɣ(α 1) ( N; α C), (5) where ψ(ζ) = 1 dɣ(ζ) (6) Ɣ(ζ) dζ is the digamma function. This formula may be easily proved by mathematical induction by using basic properties of the digamma function [, 3]. Rewriting (5)intheform d dz [zα ln z] = d z α dz ln z Ɣ(α 1) [ψ(α 1) ψ(α 1)]zα Ɣ(α 1) ( N; α C) (7) and using (7) in(4), after straightforward manipulations we obtain R n (z) = P n (z) ln z 1 1 n d n (z 1) n d (z 1) n ln z 1 n 1 n! dz n dz z 1 n n z 1 [ψ(n 1) ψ(n 1)]. (8) z 1 From the Rodrigues formula P n (z) = 1 d n (z 1) n (9) n n! dz n it follows that P n (z) = 1 n d n (z 1) n d (z 1) n. (10) n n! dz n dz Hence, it is seen that the sum of the first and the second terms on the right-hand side of (8) is zero and we have z 1 n n z 1 R n (z) = [ψ(n 1) ψ(n 1)]. (11) z 1 If the summation variable is changed from to = n, use is made of one of the elementary properties of the binomial coefficient and the primes are omitted, (11) is cast into z 1 n n z 1 R n (z) = [ψ(n 1) ψ( 1)]. (1) z 1 (It is to be noted that the lower summation limit in (11) may be shifted to 1 while the upper summation limit in (1) ton 1.) The two representations of R n (z) given in (11) and (1) supplement formulae collected in section 5. in [1].
Addendum 14889 We have already mentioned that the Jolliffe s formula () may also be exploited to rederive in a simple manner formula (5.90) from [1]. To show this, we note that if the binomial theorem is used to expand (z 1) n in a sum of powers of z 1 and the result is inserted into (3), this yields R n (z) = P n (z) ln z 1 ( 1) n d n!(n )! dz n [ (z 1) n ln z 1 ]. (13) By applying formula (5), after some obvious movements the above equation becomes R n (z) = P n (z) ln z 1 ( 1) n (n )! z 1 ln z 1 (!) (n )! ( 1) n (n )! z 1 [ψ(n 1) ψ( 1)]. (14) (!) (n )! By virtue of the Murphy s formula P n (z) = ( 1) n (n )! z 1, (15) (!) (n )! we see that the first and the second terms on the right-hand side of (14) cancel out. This yields R n (z) = ( 1) n (n )! z 1 [ψ(n 1) ψ( 1)] (16) (!) (n )! which is identical with equation (5.90) in [1]. Having established the representations (11) and (1) ofr n (z), we may obtain two expressions for the Legendre function of the second ind of integer degree, Q n (z). To show this, we recall (cf [1, sections 5.3 and 6.1]) that the Christoffel polynomial W n 1 (z) appearing in the equation Q n (z) = 1 P n(z) ln z 1 z 1 W n 1(z) (z C \ [ 1, 1]) (17) is related to the polynomial R n (z) through W n 1 (z) = 1 [R n(z) ( 1) n R n ( z)]. (18) Using R n (z) in the form (11) and employing (1) to evaluate R n ( z) gives z 1 n n z 1 W n 1 (z) = [ψ(n 1) ψ( 1)]. (19) z 1 If the roles played by (11) and (1) are interchanged, this results in z 1 n n z 1 W n 1 (z) = [ψ( 1) ψ(n 1)]. (0) z 1 Inserting (19) and (0) into(17) leads to the following representations of Q n (z): Q n (z) = 1 P n(z) ln z 1 z ± 1 n n ( z 1 z 1 [ψ(n 1) ψ( 1)] z ± 1 (z C \ [ 1, 1]). (1) )
14890 Addendum These representations of Q n (z) are seen to be direct counterparts of the following representations of P n (z) [4, p 17]: z ± 1 n ( P n (z) = F 1 n, n; 1; z 1 ) z ± 1 n = z ± 1 ( n ) ( z 1 z ± 1 ). () For z = x [ 1, 1], the Legendre function of the second ind of integer degree may be expressed as Q n (x) = 1 P n(x) ln 1x 1 x W n 1(x) ( 1 x 1), (3) where W n 1 (x) = W n 1 (z = x). (4) Adopting the common parametrization x = cos θ (0 θ π) (5) from (3), (4), (19) and (0) we obtain Q n (cos θ) = P n (cos θ)ln cot θ and cos n θ n ( 1) [ψ(n 1) ψ( 1)]tan θ (6) Q n (cos θ) = P n (cos θ)ln cot θ sin n θ n ( 1) n [ψ(n 1) ψ( 1)] cot θ. (7) These representations of Q n (cos θ)harmonize with the following nown [4, p 18] expressions for P n (cos θ): P n (cos θ) = cos n θ ( F 1 n, n; 1; tan θ ) and = cos n θ n ( 1) tan θ P n (cos θ) = ( 1) n sin n θ ( F 1 n, n; 1; cot θ ) (8) = sin n θ ( n ( 1) n ) cot θ, (9) respectively. We are not aware of any appearance of either of formulae (11), (1), (1), (6)or(7)in the literature.
Addendum 14891 References [1] Szmytowsi R 006 On the derivative of the Legendre function of the first ind with respect to its degree J. Phys. A: Math. Gen. 39 15147 Szmytowsi R 007 J. Phys. A: Math. Theor. 40 7819 (corrigendum) [] Magnus W, Oberhettinger F and Soni R P 1966 Formulas and Theorems for the Special Functions of Mathematical Physics 3rd edn (Berlin: Springer) [3] Gradshteyn ISand Ryzhi IM1994 Table of Integrals, Series, and Products 5th edn (San Diego, CA: Academic) [4] Robin L 1957 Fonctions Sphériques de Legendre et Fonctions Sphéroïdales vol 1 (Paris: Gauthier-Villars)