On Integrals Involving Universal Associated Legendre Polynomials and Powers of the Factor (1 x 2 ) and Their Byproducts

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1 Commun. Theor. Phys. 66 (216) Vo. 66, No. 4, October 1, 216 On Integras Invoving Universa Associated Legendre Poynomias and Powers of the Factor (1 x 2 ) and Their Byproducts Dong-Sheng Sun ( 孙东升 ), 1, Yuan You ( 尤源 ), 1 Fa-Lin Lu ( 陆法林 ), 1 Chang-Yuan Chen ( 陈昌远 ), 1 and Shi-Hai Dong ( 董世海 ) 2, 1 Schoo of New Energy and Eectronics, Yancheng Teachers University, Yancheng 2242, China 2 CIDETEC, Instituto Poitécnico Naciona, Adofo Lpez Mateos, CDMX, C.P. 77, Mexico (Received August 2, 216; revised manuscript received August 18, 216) Abstract The associated Legendre poynomias pay an important roe in the centra fieds, but in the case of the non-centra fied we have to introduce the universa associated Legendre poynomias (x) when studying the modified Psch-Teer potentia and the singe ring-shaped potentia. We present the evauations of the integras invoving the universa associated Legendre poynomias and the factor (1 x 2 ) p as we as some important byproducts of this integra which are usefu in deriving the matrix eements in spin-orbit interaction. The cacuations are obtained systematicay using some properties of the generaized hypergeometric series. PACS numbers: 2.3.Gp, 2.3.-f Key words: universa associated-legendre poynomias, generaized hypergeometric series, parity 1 Introduction As we know, the associated Legendre poynomias pay an important roe in the centra fieds when one soves the physica probems in the spherica coordinates. Nevertheess, in the case of the non-centra fied we have to introduce the universa associated Legendre poynomias (x) when studying the modified Psch Teer[1 3] and the singe ring-shaped potentia, [42] in which the quantum numbers and m are not integer, but their difference m n is integer. Its series expression is [( m )/2] (x) (1 x2 ) m /2 () ν Γ(2 2ν + 1) 2 ν!( m 2ν)!Γ( ν + 1) x m 2ν, (1) ν whose properties have been studied in Refs. [1, 3, 13]. For exampe, using one of its properties (2 + 1)x (x) ( m + 1) +1(x) + ( + m ) (x), (2) as we as its normaization reations, we have obtained an important resut which is unavaiabe in popuar textbooks and handbooks in Ref. [14] +1 x 2 (x)]2 dx 2Γ( + m + 1)[2 ( + 1) 2(m ) 2 1] (2 1)(2 + 1)(2. (3) + 3)n! Other reevant important integras have aso been obtained since they are unavaiabe in cassic handbooks. [156] In our recent study, [17] we have obtained some important integras, so as I ± A (a, τ) x a (x)]2 /(1 ± x) τ dx, a, 1, 2, 3, 4, 5, 6, τ 1, 2, 3, I B (b, σ) x b (x)]2 /(1 x 2 ) σ dx, b, 2, 4, 6, 8, σ 1, 2, 3, I ± C (c, κ) x c (x)]2 /(1 x 2 ) κ (1 ± x)dx, c, 1, 2, 3, 4, 5, 6, 7, 8, κ 1, 2, which are unavaiabe in cassic handbooks. We find that a of these integras depend on the integras (x)]2 /(1 + x) τ dx, τ 1, 2, 3. However, when the parameters are taken as arbitrary positive rea numbers, the integras become rather difficut to evauate. In this work, our main aim is to attack this probem. Supported by the Nationa Natura Science Foundation of China under Grant No and Partiay by SIP-IPN, Mexico E-mai: yctcsds@126.com Corresponding author, E-mai: dongsh2@yahoo.com c 216 Chinese Physica Society and IOP Pubishing Ltd

2 37 Communications in Theoretica Physics Vo. 66 That is, the integras to be cacuated are defined as L A (, m ; a, p) L ± B (, m ; b, p) L ± C (, m ; c, p) (x)]2 /(1 x 2 ) p+1 dx, x 2a (x)]2 /(1 x 2 ) p+1 dx, x b (x)]2 /(1 ± x) p dx, x c (x)]2 /[(1 x 2 ) κ (1 ± x)]dx, respectivey. The parameters a, b, c, p and κ are arbitrary positive rea numbers. It shoud be noted that a definite integras presented here for integer quantum numbers and m in (x) are aso usefu since they are unavaiabe in cassica handbooks of integras. The rest of this work is organized as foows. In Sec. 2 we derive the definite integras L(, m ; p) and L A (, m ; a, p). The definite integras L ± B (, m ; b, p) and L ± C (, m ; c, p) are derived in Secs. 3 and 4, respectivey. Some concuding remarks are given in Sec Definite Integras L(, m ; p) and L A (, m ; a, p) To sove this probem, et us begin by considering the reation between the (x) and the hypergeometric functions [13] m (x) N (1 x2 ) m /2 ( 2 F 1 n, + m + 1; m + 1; 1 x 2 ), N m Γ( + m + 1) 2 m n!γ(m + 1), n m, 1, 2,... (4) Now, et us first study the integra (x)]2 (1 x 2 ) p+1 dx (1 x 2 ) m p ( [2F 1 n, +m +1; m +1; 1 x ) dx, (5) 2 which is hepfu in deriving other integras. Using important identities for the hypergeometric series Refs. [15, 18], i.e., 2F 1 [a, b; (a + b + 1)/2; x] 2 F 1 [a/2, b/2; (a + b + 1)/2; 4x(1 x)], (6) [ 2 F 1 (a, b; a + b + 1/2; x) we have 3 F 2 (2a, 2b; a + b; 2a + 2b, a + b + 1/2; x), (7) L(, m ; p) j (1 x 2 ) m p 3 F 2 [ n, 2m + n + 1, m + 1/2, 2m + 1, m + 1; (1 x 2 )]dx ( n) j (2m + n + 1) j (m + 1/2) j (2m + 1) j (m + 1) j j! According to the definition of the beta function B(x, y) Take a new variabe ν 2t 1, the above beta function is modified as B(x, y) 1 2 x+y (1 x 2 ) j+m p dx. (8) t x (1 t) y dt, (Re x >, Re y > ). (9) (1 + ν) x (1 ν) y dν, (Re x >, Re y > ). (1) Based on this, the reation B(x, y) Γ(x)Γ(y)/Γ(x + y) and Γ(2x) 2 2x Γ(x)Γ(x + 1/2)/ π, we have πγ(j p + m (1 x 2 ) j+m p ) dx Γ(1/2 + j p + m ), Re (j p + m ) >. (11) Thus, the integra (8) can be simpified as L(, m + m + 1) πγ(m p) ; p) 2 m n!γ(m + 1) Γ(m p + 1/2) 4 F 3 ( n, 2m + n + 1, m + 1/2, m p; 2m + 1, m + 1, m p + 1/2; 1), m > p. (12) Now, we give the resuts for some specia cases n m, p, and p. [ Γ(2m + 1) πγ(m p) 2 m Γ(m + 1) Γ(m p + 1/2) 3 F 2 (, m + 1/2, m p; m + 1, m p + 1/2; 1) 4m [Γ(m + 1/2) Γ(m p) πγ(m p + 1/2) L(, m ; ) + m + 1) πγ(m ) 2 m n!γ(m + 1) Γ(m + 1/2), m > p, (13) 3 F 2 ( n, 2m + n + 1, m ; 2m + 1, m + 1; 1)

3 No. 4 Communications in Theoretica Physics 371 Γ( + m + 1) m, (14) n! L(, m + m + 1) πγ(m + 1) ; ) 2 m n!γ(m + 1) Γ(m + 3/2) 3 F 2 ( n, 2m + n + 1, m + 1/2; 2m + 1, m + 3/2; 1) 2 Γ( + m + 1) 2. (15) + 1 n! We find that some of these specia resuts are essentiay same as those given in Refs. [17, 19 22] when and m are taken integers. Let us study the foowing integra based on the resut (12) L A (, m x 2a ; a, p) (x)]2 (1 x 2 dx. (16) ) p+1 Simiary if we define t s 2 in Eq. (9), then the beta function can be rewritten as B(x, y) 2 s 2x (1 s 2 ) y ds, (Re x >, Re y > ). (17) Using this and aso the reation between the beta function and the Gamma function, we obtain the foowing integra x 2a (1 x 2 ) j+m p dx [1 + () 2a ] x 2a (1 x 2 ) j+m p dx [1 + ()2a ] Γ(a + 1/2)Γ(m + j p) 2 Γ(a + m + j p + 1/2). (18) Thus, the integra (16) can be simpified as L A (, m ; a, p) a > /2, m p >. + m + 1) Γ(a + 1/2)Γ(m p) [1 + () 2a ] 2 m n!γ(m + 1) Γ(a + m p + 1/2) 2 4 F 3 ( n, 2m + n + 1, m + 1/2, m p; 2m + 1, m + 1, a + m p + 1/2; 1), (19) 3 Definite Integra L ± B (, m ; b, p) Let us study another type of integra in terms of the resut (8) which can be rewritten as L + B (, m ; b, p) j Making use of the foowing reation [15,18] 2F 1 (α, β; γ; x) the integra in Eq. (21) can be evauated as x b (1 x 2 ) m +j (1 + x) p dx L ± B (, m ; b, p) x b (x)]2 (1 ± x) p dx, (2) x b (1 x 2 ) m (1 + x) p 3F 2 ( n, m + + 1, m + 1/2; 2m + 1, m + 1; (1 x 2 ))dx ( n) j (m + + 1) j (m + 1/2) j (2m + 1) j (m + 1) j j! Γ(γ) Γ(β)Γ(γ β) x b (1 x 2 ) m +j (1 + x) p dx. (21) t β (1 t) γ β (1 x t) α dt, (22) x b (1 + x) m +j p (1 x) m +j dx + () b x b (1 x) m +j p (1 + x) m +j dx Γ(b + 1)Γ(m + j + 1) Γ(m 2F + j + b + 2) 1 (p m j, b + 1; m + j + b + 2; ) + () b Γ(b + 1)Γ(m + j p + 1) Γ(m + j + b p + 2) 2F 1 ( m j, b + 1; m + j + b p + 2; ). (23) Obviousy, the vaue of parameter b determines the fina resut of this integra. Substitution of this into Eq. (21) yieds L + B (, m + m + 1) n ( n) j (2m + n + 1) j (m + 1/2) j ; b, p) 2 m n!γ(m + 1) (2m + 1) j (m Γ(b + 1) + 1) j j! j [ Γ(m + j + 1) Γ(m + j + b + 2) 2 F 1 (p m j, b + 1; m + j + b + 2; ) + () b Γ(m + j p + 1) Γ(m + j + b p + 2) 2 F 1 ( m j, b + 1; m + j + b p + 2; ) ], (24)

4 372 Communications in Theoretica Physics Vo. 66 where b >, m p >. As far as the integra L B (, m ; b, p) corresponding to the integra invoving the factor 1/(1 x) p, the foowing reation L B (, m ; b, p) () b L + B (, m ; b, p) is satisfied. In the specia case b, integra (23) reduces to (1 x 2 ) m +j (1 + x) p dx 2 F 1 (p m j, 1; m + j + 2; ) m + 2 F 1 ( m j, 1; m + j p + 2; ) + j + 1 m. (25) + j p + 1 Moreover, based on the beta function, we have (1 x 2 ) m +j (1 + x) p dx (1 + x) m +j p (1 x) m +j dx 2 2m +2j p+1 B(m + j p + 1, m + j + 1) πγ(m p + 1)Γ(m + 1)(m p + 1) j (m + 1) j Γ(m p/2 + 1)Γ(m p/2 + 3/2)(m p/2 + 1) j (m p/2 + 3/2) j. (26) In addition, for b Eq. (24) is simpified as L + B (, m + m + 1) πγ(m p + 1)Γ(m + 1) ;, p) 2 m n!γ(m + 1) Γ(m p/2 + 1)Γ(m p/2 + 3/2) m p >. 4 F 3 ( n, 2m + n + 1, m + 1/2, m p + 1; 2m + 1, m + 1 p/2, m + 3/2 p/2; 1), 4 Definite Integra L ± C (, m ; c, p) Finay, we are going to study the third integra L ± C (, m ; c, κ) L + C (, m ; c, κ) x c (x)]2 (1 x 2 ) κ dx, (28) (1 ± x) j x c (1 x 2 ) m (1 x 2 ) κ (1 + x) 3 F 2 ( n, 2m + n + 1, m + 1/2; 2m + 1, m + 1; (1 x 2 ))dx ( n) j (2m + n + 1) j (m + 1/2) j (2m + 1) j (m + 1) j j! x c (1 x 2 ) m +j κ dx (1 + x) + m + 1) m ( n) j (2m + n + 1) j (m + 1/2) j Γ(c + 1) 2 m n!γ(m + 1) (2m + 1) j j (m + 1) j j! [ Γ(m + j κ + 1) Γ(m + j κ + c + 2) 2 F 1 (κ m j + 1, c + 1; m + j κ + c + 2; ) + () c Γ(m + j k) Γ(m + j + c κ + 1) 2 F 1 (κ m j, c + 1; m + j κ + c + 1; ) c >, m κ >. (29) ] (27) Likewise, the integra L C (, m ; c, κ) corresponding to the integra invoving the factor 1/(1 x) has the property L C (, m ; c, κ) () c L + C (, m ; c, κ). 5 Concuding Remarks We have cacuated an important integra (x)]2 /(1 x 2 ) p+1 dx using the basic properties of the generaized hypergeometric function 4 F 3 (a, b, c, d; e, f, g; 1) and then systematicay obtained other interesting integras L A (, m ; a, p), L + B (, m ; b, p) and L + C (, m ; c, κ) given in Eqs. (19), (24), and (29), respectivey. The integras L B (, m ; b, p) and L C (, m ; c, κ) can be derived easiy by parity. What have we done is the generaization of our previous resuts for the integras whose parameters are taken as natura numbers [17] to those for arbitrary rea numbers. Such resuts wi be more usefu and hepfu in practice probems. It shoud be emphasized that these integras cannot be found in the popuar integra handbooks. The resuts presented here are interesting both in mathematics itsef and in theoretica physics. On the other hand, the resuts for integers m and are aso important since a of them are unavaiabe in popuar handbooks besides they are usefu in cacuating the matrix eements in spin-orbit interactions and other reevant probems. Acknowedgments We woud ike to thank the referee for making positive and invauabe suggestions, which have improved the manuscript.

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