Some integrals and series involving the Gegenbauer polynomials and the Legendre functions on the cut ( 1, 1)

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1 Integral Transforms and Special Functions Vol. 3, No. 11, November 01, Some integrals and series involving the Gegenbauer polynomials and the Legendre functions on the cut, 1) Radosław Szmytkowski* Atomic Physics Division, Department of Atomic Physics and Luminescence, Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Narutowicza 11/1, PL Gdańsk, Poland Received 14 July 011; final version received 16 November 011) In this study, we use the recent findings of Cohl [On a generalization of the generating function for Gegenbauer polynomials, arxiv: v1] and evaluate two integrals involving the Gegenbauer polynomials: x dt1 t ) / x t) κ/ Cn t) and x dt1 t ) / t x) κ/ Cn t), both with Re > 1,Reκ<1, < x < 1. The results are expressed in terms of the on-the-cut associated Legendre functions Pn+/ κ ±x) and Qκ n+/ x). In addition, we find closed-form representations of the series ±) n [)/]Pn+/ κ ±x)c n t) and ±) n [)/]Qn+/ κ ±x)c n t), both with Re > 1,Reκ<1, < t < 1, < x < 1. Keywords: special functions; Legendre functions; Gegenbauer polynomials; Fourier expansions MSC 010: 33C55; 33C45; 33C05 1. Introduction Recently, Cohl [1] has derived the integral formula dt 1 t ) / C n t) z t) κ+1/ π)ɣn + ) e iπ κ) 3/ n!ɣ + 1)Ɣκ + 1/) z 1) κ)/ Q κ n+/ z) Re > 1 ), κ C, z C \,1], 1.1) * radek@mif.pg.gda.pl ISSN print/issn online 01 Taylor & Francis

2 848 R. Szmytkowski where Cn t) is the Gegenbauer polynomial, while Qμ ν z) is the associated Legendre function of the second kind. The integral 1.1) generalizes Gormley s result [4]: dt 1 t ) / C n t) z t π)ɣn + ) e iπ/) z 1) /)/ Q 1/ 3/ n+/ n!ɣ + 1) z) Re > 1 ), z C \,1], 1.) which, in turn, is an extension of the celebrated Neumann s integral formula [6]: dt P nt) z t Q nz) z C \[, 1]), 1.3) where P n t) is the Legendre polynomial. The factor )/ appearing in front of the integrals in Equations 1.1) and 1.) and also at some other places in the text) is only seemingly awkward and has been introduced to avoid the difficulty one otherwise encounters when 0. From Equation 1.1) and from the closure relation for the Gegenbauer polynomials, which is Ɣ ) π n!) δt t ) Ɣn + ) C n t)c n t ) 1 t ) /)/ 1 t ) /)/ Re > 1 ), < t, t < 1 here δt t ) is the Dirac delta function), one may deduce the summation formula [1] Qκ πɣκ+ 1/) n+/ z)c n t) eiπκ ) +1/ Ɣ + 1) z 1) κ )/ z t) κ+1/ 1.4) Re > 1, κ C, < t < 1, z C \,1] ). 1.5) The particular case of this relation with κ has been known before [5, p. 183]. For κ 1/, Equation 1.5) reduces to Heine s identity: n + 1)Q n z)p n t) 1 z t < t < 1, z C \,1]). 1.6) In Section, we show that one may use the relation in Equation 1.1) to evaluate some further definite integrals involving the Gegenbauer polynomials. Furthermore, in Section 3, we exploit the identity 1.5) to determine closed-form representations of some series involving the Gegenbauer polynomials and the associated Legendre functions on the cut. While particular cases of the relations that we arrive at in the present work, corresponding to specific choices of the parameters and κ, may be found in the literature on special functions, we are not aware of any appearance of these relations in their most general forms derived below. Throughout this paper, it is understood that z 1) α z 1) α z + 1) α argz ± 1) <π). 1.7)

3 Integral Transforms and Special Functions 849. Evaluation of some definite integrals involving the Gegenbauer polynomials Let us assume that Re κ< 1. We proceed to the investigation of the limit of the formula in Equation 1.1) asz x ± i0, with < x < 1. Exploiting the identities x ± i x, x ± i0 1 e ±iπ 1 x) < x < 1).1) and e iπμ Q μ ν x ± i0) e±iπμ/ [ Q μ ν x) iπ Pμ ν x) ] < x < 1),.) where Pν μx) and Qμ ν x) are the associated Legendre functions on the cut, we obtain dt 1 t ) / Cn t) π)ɣn + ) x ± i0 t) κ+1/ 3/ n!ɣ + 1)Ɣκ + 1/) [ 1 x ) κ)/ Q κ n+/ x) iπ ] Pκ n+/ x) Re > 1 ),Reκ<1, < x < 1..3) The reason for imposing the above constraint on κ is now clear: we have had to exclude a non-integrable singularity at t x that otherwise occurs.) Since it holds that x t t < x), x ± i0 t e ±iπ.4) t x) t > x), we may split the integrals appearing on the left-hand side of Equation.3) as follows: dt 1 t ) / Cn t) x dt 1 t ) / Cn t) x ± i0 t) κ+1/ x t) κ+1/ + e iπκ+1/) dt 1 t ) / Cn t) x t x) κ+1/ Re > 1 ),Reκ<1, < x < 1..5) If we insert Equation.5) into Equation.3) and then separately equate the terms corresponding to the choices of the upper and the lower signs on both sides of the resulting equation, we arrive at an inhomogeneous algebraic system for the two integrals standing on the right-hand side of Equation.5). Solving this system and using the well-known identity we obtain x dt 1 t ) / C n t) t x) κ+1/ Ɣζ)Ɣ1 ζ) π sinπζ ),.6) π)ɣn + )Ɣ1/ κ) / n!ɣ + 1) Re > 1 ),Reκ<1, < x < 1 1 x ) κ)/ P κ n+/ x).7)

4 850 R. Szmytkowski and x dt 1 t ) / C n t) x t) κ+1/ π)ɣn + ) 3/ n!ɣ + 1)Ɣκ + 1/) 1 x ) κ)/ Q κ n+/ x) π Pκ n+/ [π x) cot κ + 1 )]} Re > 1,Reκ<1, < x < 1 ),.8) respectively. The right-hand side of the latter equation may be simplified considerably after one exploits the known relation 1 Pν μ x) Pμ ν x) cos[πν + μ)] π Qμ ν x) sin[πν + μ)] < x < 1).9) and the identity.6). This yields x dt 1 t ) / C n t) x t) κ+1/ π)ɣn + )Ɣ1/ κ) ) n 1 x ) κ)/ P κ / n+/ n!ɣ + 1) x) Re > 1 ),Reκ<1, < x < 1..10) Equation.10) may be also derived from Equation.7) after one makes the simultaneous replacements t t and x x in the latter and subsequently exploits the property Cn t) )n Cn t). Equations.7) and.10) constitute the result of this section. Their particular cases, corresponding to making the choices 1 and κ 0, may be found, albeit with the right-hand sides written in less compact forms, in [5, p. 61] and [3, p. 187]. 3. Evaluation of closed forms of some series involving the Gegenbauer polynomials and the associated Legendre functions on the cut Next, we shall draw inferences from the expansion 1.5), approaching the limit z x ± i0, with < x < 1. As in the preceding section, we assume that Re κ< 1 recall that the formula in Equation 1.5) is a consequence of Equation 1.1), which in the limit discussed here loses its sense unless the above restriction is imposed on κ). With the use of Equations.1),.) and.4), we arrive at [ Q κ n+/ x) iπ ] Pκ n+/ x) Cn t) πɣκ+ 1/) +1/ Ɣ + 1) 1 x ) κ )/ Re > 1,Reκ<1 x t) κ/ < t < x < 1), e iπκ+1/) t x) κ/ < x < t < 1) ). 3.1)

5 Integral Transforms and Special Functions 851 Hence, subtracting or adding the two relations embodied in Equation 3.1), we deduce that Pκ n+/ x)c n t) π / Ɣ + 1)Ɣ1/ κ) 1 x ) κ )/ Re > 1,Reκ<1 ) 0 < t < x < 1), t x) κ/ < x < t < 1) 3.) and Qκ n+/ x)c n t) πɣκ + 1/) x t) κ/ +1/ Ɣ + 1) 1 x ) κ )/ [ < t < x < 1), t x) κ/ cos π κ + 1 )] < x < t < 1) Re > 1 ),Reκ<1. 3.3) If in Equations 3.) and 3.3) we make the simultaneous replacements t t and x x,we obtain two further summation formulas: ) n Pκ n+/ x)c n t) π / Ɣ + 1)Ɣ1/ κ) 1 x ) κ )/ Re > 1,Reκ<1 x t) κ/ < t < x < 1), 0 < x < t < 1) ), 3.4) ) n Qκ n+/ x)c n t) [ πɣκ + 1/) +1/ Ɣ + 1) 1 x ) κ )/ x t) κ/ cos π κ + 1 )] < t < x < 1), t x) κ/ < x < t < 1) Re > 1 ),Reκ<1. 3.5) Some particular cases of the expansions 3.1) 3.5), corresponding to specific choices of κ and/or, may be found in [5, pp ], [, p. 166] and [7, pp ]. Acknowledgements I thank Dr Howard S. Cohl for drawing my attention to his work [1] and for a stimulating correspondence on the Legendre functions and related subjects.

6 85 R. Szmytkowski Notes 1. It is stated in [5, p. 170] and [, p. 144] that the domain of validity of the relation displayed in Equation.9) of this paper, and also of the counterpart expression for Qν μ x) in terms of Pν μ x) and Qν μ x), is0< x < 1. However, it is not difficult to show that if both relations hold on that interval, they must be valid for < x 0 as well.. In the second series in Section in [5, p. 18], P μ m/ cos ϑ) should be replaced by Pμ m/ cos ϑ). In the first formula in [5, p. 183], the constraint x < cos ϕ should be replaced by < cos ϕ<x < 1. References [1] H.S. Cohl, On a generalization of the generating function for Gegenbauer polynomials, arxiv: v1. [] A. Erdélyi ed.), Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, [3] A. Erdélyi ed.), Higher Transcendental Functions, Vol., McGraw-Hill, New York, [4] P.G. Gormley, A generalization of Neumann s formula for Q n z), J. London Math. Soc ), pp [5] W. Magnus, F. Oberhettinger, and R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed., Springer, Berlin, [6] F. Neumann, Entwicklung der in elliptischen Coordinaten ausgedrückten reciproken Entfernung zweier Puncte in Reihen, welche nach den Laplace schen Y n) fortschreiten; und Anwendung dieser Reihen zur Bestimmung des magnetischen Zustandes eines Rotations-Ellipsoïds, welcher durch vertheilende Kräfte erregt ist, J. Reine Angew. Math. Crelle s J.) ), pp [reprinted in: Franz Neumanns gesammelte Werke. Dritter Band, Teubner, Leipzig, 191, pp ]. In the original printing of this paper, Neumann s given-name initial was typed incorrectly as J instead of F. [7] A.P. Prudnikov,Yu.A. Brychkov, and O.I. Marichev, Integrals and Series, Volume 1: Special Functions. Supplementary Chapters, nd ed., Fizmatlit, Moscow, 003 in Russian).