Global Journal of Science Frontier Research Mathematics and Decision Sciences Volume 13 Issue 10 Version 1.0 Year 2013 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896 Computation of Some Wonderful Results Involving Certain Polynomials By Salahuddin & R.K. Khola Mewar University, India Abstract- In this paper we have developed some indefinite integrals involving certain polynomials in the form of Hypergeometric function. The results represent here are assume to be new. Keywords: hypergeometric function, lucas polynomials, gegenbaur polynomials, harmonic number, bernoulli polynomial, hermite polynomials. GJSFR-F Classification : MSC 2010 11B39, 11B68, 33C05, 33C45, 33D50, 33D60 Computation of Some Wonderful Results Involving Certain Polynomials Strictly as per the compliance and regulations of : © 2013. Salahuddin & R.K. Khola. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Notes Computation of Some Wonderful Results 13 0 Involving Certain Polynomials 2 r ea Salahuddin α & R.K. Khola σ Y 29 Abstract- In this paper we have developed some indefinite integrals involving certain polynomials in the form of Hypergeometric function. The results represent here are assume to be new. Keywords: hypergeometric function, lucas polynomials, gegenbaur polynomials, harmonic number, bernoulli polynomial, hermite polynomials. V I. Introduction X ue ersion I s a) We have the generalized Gaussian hypergeometric function of one variable s I ሺ௔భሻೖሺ௔మሻೖǥǤǤሺ௔ಲሻೖ௭ AFB(a1,a2,…,aA;b1,b2,…bB;z ) =σ௞ୀ଴ (1) XIII ሺ௕భሻೖሺ௕మሻೖǥǤǤሺ௕ಳሻೖ௞Ǩ where the parameters b1 , b2 , ….,bB are neither zero nor negative integers and A , B are non negative integers. ) F ) The series converges for all finite z if A B, converges for ŇzŇ<1 if A=B+1, diverges for all z, z 0 if A > B+1. b) Lucas Polynomials Research Volume Frontier Science of Journal Global The Lucas polynomials are the w-polynomials obtained by setting p(x) = x and q(x) = 1 in the Lucas polynomials sequence. It is given explicitly by =−−n 2 +n ++2 n (2) Lxn () 2 [(x x 4 ) + (4xx ) ] Authors α σ : Mewar University, Gangrar, Chittorgarh (Rajasthan), India. e-mails: [email protected], [email protected] ©2013 Global Journals Inc. (US) Computation of Some Wonderful Results Involving Certain Polynomials The first few are = Lx1() x 2 =+ Lx2 () x 2 (3) =+3 Lx3 () x 3x =+42 + Lx4 () x 4 x 2 c) Generalized Harmonic Number Notes The generalized harmonic number of order n of m is given by 2013 n ()m = 1 H n m (4) Year ¦ k=1 k 30 In the limit of n →∞ the generalized harmonic number converges to the Riemann zeta function ()m = ς limHmn ( ) n→∞ (5) V X d) Bernoulli Polynomial Issue ersion I XIII ) F ) Research Volume Frontier Science of In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator. Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossing of the x-axis in the unit interval does not go up as the degree of the Global Journal polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions. Explicit formula of Bernoulli polynomials is n n! Bx()= b xk , for n0, where b are the Bernoulli numbers. nn¦ − −k k k =0 kn!!()k The generating function for the Bernoulli polynomials is © 2 013 Global Journals Inc (US) Computation of Some Wonderful Results Involving Certain Polynomials e) Gegenbauer polynomials (ܤ) In Mathematics, Gegenbauer polynomials or ultraspherical polynomials Cn (x) are orthogonal polynomials on the interval [-1,1] with respect to the weight function α −1 ¦ 2 (1 − x ) 2 . They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer. Explicitly , Notes «»n 13 «»2 ¬¼ 0 Γ−+ α ()α =−kn()nk (− 2k) 2 Cxn ()¦ (1) (2)z (7) Γ−α r k =0 ()!(2)!kn k ea Y f) Laguerre polynomials 31 V X ue ersion I s s I XIII ) F ) The Laguerre polynomials are solutions Lxn () to the Laguerre differential equation xy''+−(1 x) y '+λ y = 0 , which is a special case of the more general associated Laguerre Research Volume differential equation, defined by xy''++(ν 1− x) y' +λ y = 0 , where λ and ν are real numbers with ν =0. The Laguerre polynomials are given by the sum Frontier n − k = (1)!n k Lx() x Science n ¦ − (8) k=0 kkn!!()k! of g) Hermite polynomials Journal Global ©2013 Global Journals Inc. (US) Computation of Some Wonderful Results Involving Certain Polynomials The Hermite polynomials Hxn ( ) are set of orthogonal polynomials over the domain (-,) 2 with weighting function e− x . The Hermite polynomials Hxn ( ) can be defined by the contour integral n! −+2 −− Hz()= err21znt dt , n π v³ 2 i Where the contour incloses the origin and is traversed in a counterclockwise direction Notes (Arfken 1985, p. 416). 2013 The first few Hermite polynomials are Year = Hx0 () 1 32 = Hx1() 2 x Hx()=−4x2 2 2 (9) =−3 Hx3 () 8 x 12 x V =−+42 Hx4 ( ) 16 x48 x 12 X h) Legendre function of the first kind Issue ersion I XIII ) F ) Research Volume Frontier The Legendre polynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p. 302), are solutions to Science the Legendre differential equation. If l is an integer, they are polynomials. The Legendre of polynomials Pxn ( ) are illustrated above for and n=1, 2, ..., 5. The Legendre polynomials Pxn ( ) can be defined by the contour integral −1 1 −− P ()z =−(1 2tz+ t 21 )2 tn dt , (10) Global Journal n 2πi v³ where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416). © 2 013 Global Journals Inc (US) Computation of Some Wonderful Results Involving Certain Polynomials i) Legendre function of the second kind Notes 13 0 2 r ea Y 33 The second solution Q1( x) to the Legendre differential equation. The Legendre functions of the second kind satisfy the same recurrence relation as the Legendre polynomials. The first few are V + = 1 1 x Q0 ()x ln( ) X 2 1− x + ue ersion I x 1 x s Q ()x = ln -1 (11) s 1 ( ) I 2 1− x 2 − + XIII = 3x 1 1 x 3x Q2 ( x) ln( )- 4 1− x 2 j) Chebyshev polynomial of the first kind ) F ) Research Volume Frontier Science of Journal The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted Tn (x). They are used as an approximation to a least squares fit, and are a special case of the Gegenbauer Global = polynomial with α 0. They are also intimately connected with trigonometric multiple-angle . formulas The Chebyshev polynomial of the first kind Tn ()z can be defined by the contour integral 1 (1− t 2 )t −n−1 T() z = dt , (12) n 4πi∫ 1− 2 tz + t 2 ©2013 Global Journals Inc. (US) Computation of Some Wonderful Results Involving Certain Polynomials where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416). The first few Chebyshev polynomials of the first kind are = T0 ( x ) 1 = T1() x x =2 − T2 ( x ) 2 x 1 =3 − T3 ( x ) 4 x 3 x Notes A beautiful plot can be obtained by plotting Tn ()x radially, increasing the radius for each 2013 value of n , and filling in the areas between the curves (Trott 1999, pp. 10 and 84). Year 34 V X Issue ersion I XIII ) F ) The Chebyshev polynomials of the first kind are defined through the identity Research Volume Tn(cos θ)=cos nθ. k)Chebyshev polynomial of the second kind Frontier Science of Global Journal A modified set of Chebyshev polynomials defined by a slightly different generating function. They arise in the development of four-dimensionalspherical harmonics in angular momentum © 2 013 Global Journals Inc (US) Computation of Some Wonderful Results Involving Certain Polynomials theory. They are a special case of the Gegenbauer polynomial with α= 1. They are also intimately connected with trigonometric multiple-angle formulas. The first few Chebyshev polynomials of the second kind are = U0 ( x ) 1 U( x )= 2 x 1 (13) U( x )= 4 x2 − 1 Notes 2 =3 − U( x ) 8 x 4 x 13 3 0 2 l) Euler polynomial r ea Y 35 V X ue ersion I s s I XIII ) F ) Research Volume The Euler polynomial En ( x) is given by the Appell sequence with 1 g() t=( et + 1) , 2 Frontier giving the generating function Science xt ∞ n 2e t of ≡ . t ∑ En () x (14) e +1 = n! n 0 Journal The first few Euler polynomials are = Global E0 ( x ) 1 = − 1 E1()x x 2 E () x = x 2 − x (15) 2 3 1 E() x= x3 − x2 + 3 2 4 ©2013 Global Journals Inc. (US) Computation of Some Wonderful Results Involving Certain Polynomials m)Generalized Riemann zeta function Notes 2013 Year 36 V X Issue ersion I XIII ) The Riemann zeta function is an extremely important special function of mathematics and F ) physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem.