Computation of Some Wonderful Results Involving Certain Polynomials by Salahuddin & R.K

Home , Appell sequence, Chebyshev polynomials, Jacobi polynomials, Sheffer sequence

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

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© 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 b , b , ….,b are neither zero nor negative integers and A , B are non 1 2 B 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

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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]

The first few are = Lx1() x Lx()=+x2 2 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 k 1 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

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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

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 Γ−+ α 0 ()α =−kn()nk (− 2k) 2 Cxn ()¦ (1) (2)z (7) Γ−α r k =0 ()!(2)!kn k ea Y f) Laguerre polynomials 31

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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 (1− )!k n = k

Lx() x Science n ¦ − (8) k=0 kkn!!()k!

of g) Hermite polynomials

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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

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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).

i) Legendre function of the second kind

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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 )

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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 πi∫ − tz + t 2 4 1 2

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).

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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

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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

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

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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 ≡ E() x . (14) t + ∑ n e 1 n=0 n !

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

m)Generalized Riemann zeta function

Notes

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physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably

Research Volume the Riemann hypothesis) that remain unproved to this day. The Riemann zeta function is defined over the complex plane for one complex variable, which is conventionally denoted s

(instead of the usual ) in deference to the notation used by Riemann in his 1859 paper that

Frontier founded the study of this function (Riemann 1859).

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The plot above shows the "ridges" of for and . The fact that the ridges appear to decrease monotonically for is not a coincidence since it turns out that monotonic decrease implies the Riemann hypothesis (Zvengrowski and Saidak 2003;

Borwein and Bailey 2003, pp. 95-96).

On the real line with , the Riemann zeta function can be defined by the integral

− 1 ∞ u x 1 ς ≡ where Γ is the gamma function. ()x ∫ u du , ()x Γ(x)0 e − 1 Notes 13

n) Complex infinity 0 2

Complex infinity is an infinite number in the complex plane whose complex argument is r ea

unknown or undefined. Complex infinity may be returned byMathematica, where it is Y represented symbolically by ComplexInfinity. The Wolfram Functions Site uses the 37 notation to represent complex infinity.

II. Main Results

V L() x + 1 1 2 1 2 n 2 n ∫ dx = x 2F1( , ; ; x ) + C (17) X 1− xn 2 2 n n ue ersion I s s I ()x H + XIII 1 1 1 n 1 n ∫ dx = x 2F1( , ; ; x ) + C (18) 1− xn 2 n n

)

B() x + + F ) 1 1 1 2 n 2 n 1 1 n 1 n ∫ dx = x [x 2F1( , ; ; x ) - 2F1( , ; ; x )] + C (19) 1− xn 2 2 n n 2 n n

C() x + Research Volume 1 2 1 2 n 2 n ∫ dx = x 2F1( , ; ; x ) + C (20) 1− xn 2 n n

Frontier F() x + 1 1 2 1 2 n 2 n dx = x 2F1( , ; ; x ) + C (21) ∫ n 1− x 2 2 n n Science

of L[] x + + 1 1 1 n 1 n 1 2 1 2 n 2 n dx = x 2F1 , ; ; x x 2F1 , ; ; x + C (22) ∫ ( ) - ( ) Journal 1− xn 2 n n 2 2 n n

H() x + Global 1 2 1 2 n 2 n dx = x 2F1( , ; ; x ) + C (23) ∫ n 1− x 2 n n

P() x + 1 1 2 1 2 n 2 n ∫ dx = x 2F1( , ; ; x ) + C (24) 1− xn 2 2 n n

T() x + 1 1 2 1 2 n 2 n ∫ dx = x 2F1( , ; ; x ) + C (25) 1− xn 2 2 n n

U() x + 1 2 1 2 n 2 n ∫ dx = x 2F1( , ; ; x ) + C (26) 1− xn 2 n n

E() x 1 1 2 n + 2 1 1 n +1 otes 1 dx = x x F , ; ; x n F , ; ; x n + C (27) N [ 2 1( ) - 2 1( )] ∫ n − x 2 2 n n 2 n n 2013 1

ζ Year (1,)x = ∞ +C (28) ∫ n 38 1− x

References Références Referencias

V 1. Abramowitz, Milton; Stegun, Irene A.(1965), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover. X 2. Andrews, L.C.(1992) ; Special Function of mathematics for Engineers,Second Edition, McGraw-Hill Co Inc., New York. Issue ersion I 3. Arfken, G. ; Mathematical Methods for Physicists, Orlando, FL: Academic Press.

XIII 4. Bells, Richard, Wong, Roderick ; Special Functions , A Graduate Tex t, Cambridge Studies in Advanced Mathematics, 2010. 5. Prudnikov, A. P., Marichev, O. I., and Brychkov, Yu. A. ; Integrals and Series, Vol. 3:

) More Special Functions., NJ:Gordon and Breach, 1990.

F )

6. Rivlin, T. J. ;Chebyshev Polynomials., New York: Wiley, 1990. 7. Wikipedia(2013,October), Bernoulli Polynomials. Retrived from website: http://en.wikipedia.org/wiki/Bernoulli_polynomials.

Research Volume 8. Wikipedia(2013,October), Gegenbaur Polynomials. Retrived from website: http://en.wikipedia.org/wiki/Gegenbauer_polynomials.

Frontier 9. WolframMathworld(2013,October), Riemann Zeta function. Retrived from website: http://mathworld.wolfram.com/RiemannZetaFunction.html. 10. WolframMathworld(2013,October), Harmonic Number . Retrived from website: Science http://mathworld.wolfram.com/HarmonicNumber.html.

of 11. WolframMathworld(2013,October), LaguerrePolynomials. Retrived from website: http://mathworld.wolfram.com/LaguerrePolynomial.html. 12. WolframMathworld(2013,October), Hermite Polynomials. Retrived from website: http://mathworld.wolfram.com/HermitePolynomial.html.

Global Journal 13. WolframMathworld(2013,October), Legendre Polynomial. Retrived from website: http://mathworld.wolfram.com/LegendrePolynomial.html. 14. WolframMathworld(2013,October), Chebyshev Polynomial of the first kind. Retrived from website: http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html. 15. WolframMathworld(2013,October), Lucas polynomials. Retrived from website: http://mathworld.wolfram.com/LucasPolynomials.html.