Certain Indefinite Integrals Involving Lucas Polynomials and Harmonic

Home , Harmonic number

Global Journal of Science Frontier Research Mathematics and Decision Sciences Volume 12 Issue 11 Version 1.0 Year 2012 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896

Certain Indefinite Integrals Involving Lucas Polynomials and Harmonic Number By Salahuddin P.D.M College of Engineering, India Abstract - In this paper we have established certain indefinite integrals involving Harmonic number and Lucas Polynomials. The results represent here are assume to be new. Keywords and Phrases : Polylogarithm; Lucas polynomials; Harmonic Number; Gaussian Hypergeometric Function. GJSFR-F Classification : MSC 2010: 11B39

Certain Indefinite Integrals Involving Lucas Polynomials and Harmonic Number

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Certain Indefinite Integrals Involving Lucas Notes Polynomials and Harmonic Number 2012 r

Salahuddin ea Y

Abstract - In this paper we have established certain indefinite integrals involving Harmonic number and Lucas 37 Polynomials. The results represent here are assume to be new. Keywords and Phrases : Polylogarithm; Lucas polynomials; Harmonic Number; Gaussian Hypergeometric Function.

V I. Introduction and P reliminaries XI a) Harmonic Number th The n harmonic number is the sum of the reciprocals of the ﬁrst n natural numbers: n 1 H = (1.1) n k Xk=1

) F )

Harmonic numbers were studied in antiquity and are important in various branches of number

theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in various expressions for various special functions.

An integral representation is given by Euler Research Volume XII Issue ersion I 1 1 xn Hn = − dx (1.2) Z0 1 x

− Frontier The equality above is obvious by the simple algebraic identity below

1 xn − = 1 + x + ...... + xn (1.3) 1 x − An elegant combinatorial expression can be obtained for Hn using the simple integral transform

x = 1 u : Journal of Science − 1 1 xn 0 1 (1 u)n 1 1 (1 u)n Hn = − = − − du = − − du Global Z0 1 x − Z1 u Z0 u − 1 n = ( 1)k−1 n uk−1 du Z − k 0 Xk=1

Author : P.D.M College of Engineering, Bahadurgarh, Haryana , India. E-mail : [email protected]

n 1 = ( 1)k−1 n uk−1du − k Z Xk=1 0

n (1.4) k−1 1 n = ( 1) − k k Xk=1

b) Lucas polynomials Notes The sequence of Lucas polynomials is a sequence of polynomials deﬁned by the recurrence rela- tion 2012

r 2x0 = 2 , if n = 0 ea Y

1 L x x x , n .

n( ) =  1 = if = 1 (1 5) 1 0  x Ln (x) + x Ln (x) , if n 2 38 −1 −2 ≥  The ﬁrst few Lucas polynomials are:

V L0(x) = 2

XI L1(x) = x 2 L2(x) = x + 2

3 L3(x) = x + 3x 4 2 L4(x) = x + 4x + 2

) The ordinary generating function of the Lucas polynomials is F )

∞ n 2 xt G{Ln(x)}(t) = Ln(x)t = − . (1.6) 1 t(x + t) nX=0 −

Research Volume XII Issue ersion I c) Polylogarithm

The polylogarithm (also known as Jonquire’s function) is a special function Li (z) that is deﬁned s Frontier by the inﬁnite sum, or power series:

∞ zk Li (z) = (1.7) s ks Xk=1

It is in general not an elementary function, unlike the related logarithm function. The above definition is valid for all complex values of the order s and the argument z where z < 1.The Journal of Science | | polylogarithm is defined over a larger range of z than the above definition allows by the process of analytic continuation.

Global The special case s = 1 involves the ordinary natural logarithm (Li (z) = ln(1 z)) while the 1 − − special cases s = 2 and s = 3 are called the dilogarithm (also referred to as Spence’s function)

and trilogarithm respectively. The name of the function comes from the fact that it may alter- natively be deﬁned as the repeated integral of itself, namely that

z Li s(t) Lis+1(z) = dt (1.8) Z0 t

Thus the dilogarithm is an integral of the logarithm, and so on. For nonpositive integer orders

s, the polylogarithm is a rational function. The polylogarithm also arises in the closed form of the integral of the FermiDirac distribution and the Bose-Einstein distribution and is sometimes known as the Fermi- Dirac integral or the Bose-Einstein integral. Polylogarithms should not be confused with polylogarithmic functions nor with the oﬀset logarithmic integral which has a similar notation.

d) Generalized Gaussian Hypergeometric Function Generalized ordinary hypergeometric function of one variable is deﬁned by Notes

a1, a2, , aA ; ∞ k ··· (a1)k(a2)k (aA)kz AFB  z  = ··· 2012 (b1)k(b2)k (bB)kk! r

b1, b2, , bB ; Xk=0 ··· ea

 ···  Y or A 39 (a ) ; (a ) ; ∞ A j j=1 ((a )) zk z z A k AFB   AFB   = (1.9) ≡ B ((bB))kk! (b ) ; (b ) ; Xk=0  B   j j=1 

V b , b , , b A, B where denominator parameters B are neither zero nor negative integers and XI 1 2 ··· are non-negative integers.

II. Main Indefinite Integrals

(x) sinh x H L1(x) 1 8 6ι ι x 1 dx = e(−1− 2 )x sin Z √1 cos x −√ 1 cos x25 − 25 2 × − − ) F ) 2x 1 1 1 1 ιx ιx 1 1 3 3 ιx 2e 3F2 ι, ι, 1; ι, ι; e 2e 3F2 + ι, + ι, 1; + ι, + ι; e ×h − 2 − −2 − 2 − 2 − − 2 2 2 2 − 1 1 1 3 (2 ι)xe2x F ι, 1; ι; eιx (2 ι)xeιx F + ι, 1; + ι; eιx + − − 2 1 − 2 − 2 − − − 2 1 2 2 Research Volume XII Issue ersion I 2x 2x +(2 ι)xe 2e + Constant (2.1) − − i

(x) Frontier sin x H1 L1(x) 1 −ιx x 1 1 3 3 x dx = e (e 1) (8+6ι)3F2 ι, ι, 1; ι, ι; e Z √1 cosh x 25√1 cosh x − − 2 − 2 − 2 − 2 − − − − h 2ιx 1 1 3 3 1 3 x (8 6ι)e 3F2 +ι, +ι, 1; +ι, +ι; cosh x +sinh x +5x (2 ι)2F1 ι, 1; ι; e + − − 2 2 2 2 n − 2 − 2 −

2ιx 1 3 +(2 + ι)e 2F1 + ι, 1; + ι; cosh x + sinh x + Constant (2.2)

2 2 Journal of Science oi

(x) cos x H L1(x) 1 1 1 3 3 1 −ιx x x Global dx = e (e 1) (6 8ι)3F2 ι, ι, 1; ι, ι; e + Z √1 cosh x −25√1 cosh x − h − 2 − 2 − 2 − 2 − − − 2ιx 1 1 3 3 1 3 x +(6+8ι)e 3F2 +ι, +ι, 1; +ι, +ι; cosh x+sinh x +5x (1+2ι)2F1 ι, 1; ι; e + 2 2 2 2 n 2 − 2 − 2ιx 1 3 +(1 2ι)e 2F1 + ι, 1; + ι; cosh x + sinh x + Constant (2.3) − 2 2 oi

(x) x sin x H L (x) 2 x x 1 tan + 1 1 1 dx = cos sin π tanh−1 4 + Z √1 sin x √1 sin x 2 − 2 √2 √2 − − 1 3 ιx 3 ιx 3 ιx 4 2 4 2 4 2 + 4ιLi2 ( 1) e 4ιLi2 ( 1) e (π 2x) log 1 ( 1) e 2 − − − − − − − − − 3 ιx x x log 1 + ( 1) 4 e 2 (x 2) sin (x + 2) cos + Constant (2.4) − − − − 2 − 2

(x) cot x H L (x) 1 x x ιx ιx Notes 1 1 2 2 dx = cos sin 4ιLi2( e ) + 4ιLi2( ιe ) Z √1 sin x 2√1 sin x 2 − 2 h − − − ιx− ιx − ιx ιx ιx 2012 4ιLi2(ιe 2 ) 4ιLi2(e 2 ) ιπx + 2x log(1 e 2 ) + 2x log(1 ιe 2 ) 2x log(1 + e 2 ) r − − − − − − − ea ιx ιx ιx x + π Y 2 2 2

2x log(1 + ιe ) + 2π log(1 ιe ) + 2π log(1 + ιe ) 2π log sin

− − − 4 − 40 x + π 2π log cos + Constant (2.5) − − 4 i x H(x)L x tan 1 1( ) 1 ιx 1 1 2ιx

V ι e Li e dx = ιx 2 ( 1 + ) 4 2 1 + Z √1 sec x (−1+e ) √ 2ιx − − 2 − 2 − 8 e2ιx 1 + e p XI − 1+ q −1 ιx −2 sinh (e ) 2 2ιx 2 1 2ιx 4Li2 e log ( e ) 2 log 1 + 1 + e + − − − − 2 p 1 e2ιx e2ιx ιx e2ιx eιx −1 eιx 2 +4 log( ) log 1 + 1 + 8 log 1 + + + 4 sinh ( ) + − 2 p − p −1 ιx −1 2ιx −1 ιx −2 sinh (e ) +8ιx tanh 1 + e + 8 sinh ( e ) log 1 e

) − − p F ) 4 log( e2ιx) tanh−1 1 + e 2ιx + Constant (2.6)

− − p

(x) ιx 2 cot x H L1(x) 1 (e ι) 1 Research Volume XII Issue ersion I 1 2ιx 2ιx dx = − 1 + e 1 e Z √1 cosec x 8(eιx ι)r 1 + e2ιx − √ 1 + e2ιx − × − − − p − p 1 1 1 4Li 1 e2ιx + log2(e2ιx ) + 2 log2 1 + 1 e2ιx Frontier 2 × − 2 − 2p − 2 p − − 1 4 log 1 + 1 e2ιx log(e2ιx) + 4(2ιx log(e2ιx)) tan−1 1 + e2ιx − 2 − − − − p p −1 ιx 2ιx −2ι sin (e ) 2ιx ιx −1 ιx 2 4 1 e Li2 e + 2ιx log( 1 e + ιe ) + sin (e ) − p − p − − −1 ιx −2ι sin− 1(eιx) Journal of Science ι e e Constant . 2 sin ( ) log 1 + (2 7) − −

Global x ιx ιx ( ) − 2 − 2 tan x H1 L1(x) 1 x (1 + ι)e (1 ι)e dx = sin 8ιLi2 + 8ιLi2 − + Z √1 cos x 2√2 2 cos x 2 − √2 √2 − − x x x x (1 + ι)(cos 2 + ι sin 2 ) (1 + ι)(sin 2 ι cos 2 ) 2 +8ιLi2 + 8ιLi2 − + 2ιx 2ιπx+ − √2 √2 − ιx ιx ιx (1 ι)e− 2 (1 + ι)e− 2 (1 ι)e− 2 +4x log 1 − + 4x log 1 + 4π log 1 − − √2 √2 − − √2 −

ιx ιx − 2 − 2 (1 + ι)e − 1 2 + √2 (1 ι)e 4π log 1 + + 16 sin p log 1 − − √2 2 − √2 − ιx 2 + √2 (1 + ι)e− 2 x −1 √ 16 sin p log 1 + + 4π log 2 sin + 2 − 2 √2 2 −

(1 + ι) sin x (1 ι) cos x (1 ι) sin x (1 + ι) cos x 4x log 2 − 2 + 1 4x log − 2 + 2 + 1 − − √2 − √2 − − √2 √2 − √ √ x+π Notes −1 2 + 2 −1 ( 2 2) cot 4 2 32 sin p tanh − + ιπ + Constant (2.8) − 2 √2

2012

III. Derivation of the Integrals r ea

Y Involving the same parallel method of ref[8] , one can derive the integrals. 41 IV. Con clusion In our work we have established certain indeﬁnite integrals involving Lucas Polynomials , Har-

monic Number , and Hypergeometric function . However, one can establish such type of in-

V tegrals which are very useful for diﬀerent ﬁeld of engineering and sciences by involving these

integrals.Thus we can only hope that the development presented in this work will stimulate XI further interest and research in this important area of classical special functions.

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