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European Journal of Applied Engineering and Scientific Research, 2014, 3 (1):21-30 (http://scholarsresearchlibrary.com/archive.html)

ISSN: 2278 – 0041

Some curious results involving certain

Salahuddin and R.K. Khola

Mewar University, Gangrar, Chittorgarh (Rajasthan) , India ______

ABSTRACT

In this paper we have developed some curious results involving certain polynomials.

Key Words : , Laplace Transform, Lucas Polynomials, Gegenbaur polynomials, Harmonic number, Bernoulli , . 2010 MSC NO : 11B39, 11B68, 33C05, 33C45, 33D50, 33D60 ______

INTRODUCTION

We have the generalized Gaussian hypergeometric function of one variable

∞ ….. AFB(a 1,a 2,…,a A;b 1,b 2,…b B;z ) =∑ (1.1) ….. !

where the parameters b1 , b 2 , ….,b B are neither zero nor negative integers and A , B are non negative integers. The series converges for all finite z if A ≤ B, converges for │z│<1 if A=B +1, diverges for all z, z ≠ 0 if A > B+1.

Laplace Transform The Laplace Transform of a function f(t) , defined for all real numbers t ≥0, is the function F(s), defined by :

∞ − Fs()= Lft {()}() s = ∫ est ftdt () 0 (1.2) The parameter s is a complex number:

s=σ + i ω , with real number σ and ω .

Jacobi Polynomials In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials ) are a class of classical . They are orthogonal with respect to the weight

α β (1−x ) (1 + x ) on the interval [-1, 1]. The , and thus also the Legendre, Zernike and , are special cases of the Jacobi polynomials.

The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.

The Jacobi polynomials are defined via the hypergeometric function as follows:

21 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 201 4, 3 (1):21-30 ______

α β (α + 1) − ( , ) =n −++++α β α 1 z Pzn ()2 Fn 1 (,1 n ;1; ), n! 2 (1.3)

α + Where ( 1) n is Pochhammer’s symbol. In this case, the series for the hypergeometric function is finite, therefore one obtains the following equivalent expression:

Γ++αn Γ++++− α β (α , β ) = (n 1) n ! ( nmz 1) 1 m Pn ( z ) ∑ ( ) n!(Γ+++α β n 1)= mnm !() − ! Γ++ ( α m 1) 2 m 0 (1.4)

Lucas Polynomials

The Lucas polynomials are the w-polynomials obtained by setting p(x) = x and q(x) = 1 in the Lucas polyno mials sequence. It is given explicitly by

=−n −2 + n +2 + n Lxn ()2[( xx 4 ) + (x x 4 ) ] (1.5 ) The first few are = L1( x ) x L( x )= x 2 + 2 2 (1.6) =3 + Lx3 ( ) x 3 x =4 + 2 + Lx4 () x 4 x 2

Generalized Harmonic Number The generalized harmonic number of order n of m is given by n (m ) = 1 H n ∑ m k=1 k (1.7)

In the limit of n → ∞ the generalized harmonic number converges to the Riemann zeta function

(m ) = ς limHn ( m ) n→∞ (1.8)

22 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 2014, 3 (1):21-30 ______

Bernoulli Polynomial

In mathematics, the 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 , i.e. a 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 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( ) = bx k , for n 0, where b are the Bernoulli numbers. n∑ − n− k ≥ k k =0 k!() n k ! The for the Bernoulli polynomials is text∞ t n = B( x ) t − ∑ n e1n=0 n ! (1.9) Gegenbauer polynomials (ɑ) In Mathematics, Gegenbauer polynomials or ultraspherical polynomials C n (x) are orthogonal polynomials on the α −1 interval [-1,1] with respect to the weight function (1− x2 ) 2 . They generalize and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer . Explicitly ,

n    2  α Γ(n − k + α ) − C() () x= (1) − k (2) z ( n 2 k ) n ∑ Γα − k =0 ( )!(k n 2)! k (1.10)

23 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 2014, 3 (1):21-30 ______

Laguerre polynomials

The are solutions Ln ( x ) to the Laguerre differential equation

xy''+− (1 x ) y ' +λ y = 0 , which is a special case of the more general associated Laguerre differential equation, defined by

xy''(++−ν 1 xy )' + λ y = 0 , where λ and ν are real numbers with ν =0.

The Laguerre polynomials are given by the sum n − k = ( 1)n ! k Ln ( x ) ∑ x = k! kn !()− k ! k 0 (1.11)

Hermite polynomials

The Hermite polynomials Hn ( x ) are set of orthogonal polynomials over the domain (-∞,∞) with weighting − 2 function e x .

The Hermite polynomials Hn ( x ) can be defined by the contour integral

= n! −+r2 2 rz −− n 1 Hzn ( ) ∫ e tdt , 2πi 

Where the contour incloses the origin and is traversed in a counterclockwise direction

24 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 201 4, 3 (1):21-30 ______

(Arfken 1985, p. 416).

The first few Hermite polynomials are = H0 ( x ) 1 = H1( x ) 2 x H() x= 4 x 2 − 2 2 (1.12) =3 − Hx3 () 8 x 12 x =4 − 2 + Hx4 () 16 x 48 x 12

Legendre function of the first kind

The Legendre po lynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p. 302), are solutions to the Legendre differential equation . If l is an integer, they are polynomials . The Legendre polyn omials Pn ( x ) are illustrated above for and n=1, 2, ..., 5.

The Legendre polynomials Pn ( x ) can be defined by the contour integral −1 1 − − = − + 22 n 1 Pzn ( )∫ (1 2 tzt ) tdt , (1.13) 2πi  where the contour encloses the origin and i s traversed in a counterclockwise direction (Arfken 1985, p. 416).

Legendre function of the second kind

25 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 201 4, 3 (1):21-30 ______

The second solution to the Legendre differential equation . The Leg endre functions of the second kind satisfy the same recurrence relation as the Legendre polynomials .

The first few are 1 1+ x Q( x ) = ln( ) 0 2 1− x x 1+ x Q( x ) = ln( )-1 (1 .14) 1 2 1− x 3x2 − 1 1+ x 3x Q( x ) = ln( )- 2 4 1− x 2

Chebyshev polynomial of the first kind

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 polynomial with . 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−t2 ) t −n − 1 = , (1.15) Tn ( z ) ∫ 2 dt 4πi 1− 2 tzt + 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 − Tx3 () 4 x 3 x

A beautiful plot can be obtained by plotting Tn ( x ) radially, increasing the radius for each value of n , and filling in the areas between the curves (Trott 1999, pp. 10 and 84).

26 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 201 4, 3 (1):21-30 ______

The Chebyshev polynomials of the first kind are defined through the identity

Tn(cos θ)=cos nθ.

Chebyshev polynomial of the second kind

A modified set of Chebyshev polynomials defined by a slightly different generating function . They arise in the development of four-dimensional spherical harmonics in angular momentum theory. They are a special case of the Gegenbauer polynomial with . They are also intimately connected with trigonometric multiple-angle formulas.

The first few Cheby shev polynomials of the second kind are

= U0 ( x ) 1 U( x )= 2 x 1 (1.16) =2 − U2 () x 4 x 1 =3 − Ux3 () 8 x 4 x

27 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 2014, 3 (1):21-30 ______

Euler polynomial

The Euler polynomial En ( x ) is given by the Appell sequence with

1 g( t )= ( e t + 1) , 2 giving the generating function 2ext∞ t n ≡ E( x ) . (1.17) t + ∑ n e1n=0 n !

The first few Euler polynomials are = E0 ( x ) 1 1 E( x ) = x − 1 2 Ex( ) = x2 − x 2 (1.18) 3 1 Ex( ) = x3 − x 2 + 3 2 4

1. MAIN RESULTS

− 2 1− 2n 2 −−n− − n + xnF(2 1 ( ,;; x ) 4 x 1 − L( x ) 1 + x n dx = 2 n n + C ∫ 1 − 2(n 4) (2.1)

− 1− 1n 1 −−n− − n + nx2 F 1 (,;; x ) 2 x x 1 − H(x ) 1+ x n dx = 2 n n + C ∫ 1 − (n 2) (2.2)

− + −n 1 −1 − 2n 2 − −n ∫ B1( x ) 1 x dx = x[ (2)nnxF2 1 ( ,; ; x ) 2(n− 4)( n − 2) 2 n n

28 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 2014, 3 (1):21-30 ______

1 1n − 1 − − − − − − n +n + − ++ − + (4)nnF2 1 ( ,; ; x ) 2x 1 (2nxn 4 x 4)] C 2 n n (2.3)

− 2 1− 2n 2 −−n− − n + xnF(2 1 ( ,;; x ) 4 x 1 − C( x ) 1 + x n dx = 2 n n + C ∫ 1 − (n 4) (2.4)

− 2 1− 2n 2 −−n− − n + xnF(2 1 ( ,;; x ) 4 x 1 − F( x ) 1 + x n dx = 2 n n + C ∫ 1 − 2(n 4) (2.5)

− 1− 1n 1 −−n− − n + nx2 F 1 (,;; x ) 2 x x 1 − L[1]( s ) 1 + x n dx = 2 n n + C ∫ (x ) − (n 2) s (2.6)

− 2 1− 2n 2 −−n− − n + xnF(2 1 ( ,;; x ) 4 x 1 − P( x ) 1 + x n dx = 2 n n + C ∫ 1 − 2(n 4) (2.7)

− 2 1− 2n 2 −−n− − n + xnF(2 1 ( ,;; x ) 4 x 1 − T( x ) 1 + x n dx = 2 n n + C ∫ 1 − 2(n 4) (2.8)

− 2 1− 2n 2 −−n− − n + xnF(2 1 ( ,;; x ) 4 x 1 − U( x ) 1 + x n dx = 2 n n + C ∫ 1 − (n 4) (2.9)

− 2 1− 2n 2 −−n− − n + xnF(2 1 ( ,;; x ) 4 x 1 − H( x ) 1 + x n dx = 2 n n + C ∫ 1 − (n 4) (2.10)

− 1 1 2n − 2 − L( x ) 1 + x n dx = − x[ (2)nnxF−( ,; − ; − x n ) ∫ 1 2(n− 4)( n − 2) 2 1 2 n n

1 1n − 1 − − −2(4)nnF−( , − ; ; − x n ) +4x n + 1 (−nxn ++ 2 x − 4)] + C 2 1 2 n n (2.11)

− 1 P(x , b ) ( x ) x n + 1 dx = [xnn {2 (2− 6 n + 8) x 2 ∫ 1 12(n− 6)( n − 4)( n − 2)

133n − − 111n − − ×−−F(), ; ; xn − 3(2(( bnnnF2 −+ 10 24) () , −− ; ; x n 2 1 2 n n 2 1 2 n n

− +2xn + 1(( bn − 6)(( nx −−++− 1) 2 x 4) ( nxnx 2)(( +− 3) 2(2 x + 9))))

29 Scholars Research Library Salahuddin and R.K. Khola Euro. J. Appl. Eng. Sci. Res., 2014, 3 (1):21-30 ______− −+2 −+1 − 2n 2 − −n + (bnnnxF 3)( 812)2 1 ( , ; ; x ) ))) C (2.12) 2 n n

− 1 1 2n − 2 − E( x ) 1 + x n dx = x[ (2)nnxF−( ,; − ; − x n ) ∫ 1 2(n− 4)( n − 2) 2 1 2 n n

1 1n − 1 − − −(4)nnF−( ,; − ; − x n ) +2x n + 1 (2−nxn ++ 4 x − 4)] + C 2 1 2 n n (2.13)

where C is the integral constant.

REFERENCES

[1] Abramowitz, Milton; Stegun, Irene A.( 1965 ), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , New York: Dover. [2] Andrews, L.C.( 1992 ) ; Special Function of mathematics for Engineers ,Second Edition , McGraw-Hill Co Inc., New York. [3] Arfken, G. ; Mathematical Methods for Physicists , Orlando, FL: Academic Press. [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 . [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. [8] Wikipedia( 2013 ,October), Gegenbaur Polynomials. Retrived from website: http://en.wikipedia.org/wiki/Gegenbauer_polynomials. [9] WolframMathworld( 2013 ,October), Harmonic Number . Retrived from website: http://mathworld.wolfram.com/HarmonicNumber.html. [10] WolframMathworld( 2013 ,October), LaguerrePolynomials. Retrived from website: http://mathworld.wolfram.com/LaguerrePolynomial.html. [11] WolframMathworld( 2013 ,October), Hermite Polynomials. Retrived from website: http://mathworld.wolfram.com/HermitePolynomial.html. [12] WolframMathworld( 2013 ,October), Legendre Polynomial. Retrived from website: http://mathworld.wolfram.com/LegendrePolynomial.html. [13] WolframMathworld( 2013 ,October), Chebyshev Polynomial of the first kind. Retrived from website: http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html. [14] WolframMathworld( 2013 ,October), Lucas polynomials. Retrived from website: http://mathworld.wolfram.com/LucasPolynomials.html.

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