Certain Sequences and Its Integral Representations in Terms Of

Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 15 Issue 5 Version 1.0 Year 2015 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896

Certain Sequences and its Integral Representations in Terms of Laguerre Polynomials By Baghdadi Aloui Abstract- In this paper, we introduce a connection formula between the monomial basis and the shifted Laguerre basis. As an application, some integral representations in terms of Laguerre polynomials for certain sequences are obtained. Keywords: Laguerre polynomials, special functions, integral formulas. GJSFR-F Classification : MSC 2010: Primary 33C45; Secondary 42C05

CertainSequencesanditsIntegralRepresentationsinTermsofLaguerrePolynomials

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Notes Certain Sequences and its Integral

Representations in Terms of Laguerre 201 r ea Polynomials Y 1 Baghdadi Aloui

Abstract- In this paper, we introduce a connection formula between the monomial basis and the shifted Laguerre basis. V As an application, some integral representations in terms of Laguerre polynomials for certain sequences are obtained.

V Keywords: Laguerre polynomials, special functions, integral formulas.

ue ersion I s s I. Introduction and Main Results I

By using some special functions and some particular integrals, we recall some integral rep- XV resentations for certain integer (or real) sequences. a) Some special functions )

The Gamma function isdeﬁned by the deﬁnite integral F )

Z +∞ Γ(z) = xz−1e−x dx, 0. 0 Research Volume We can see directly, that Γ(1) = 1, and using integration by parts, that Γ(z + 1) = zΓ(z). Notice that, for z = n ∈ \{0}, the following formulas hold N n! = Γ(n + 1) Frontier Z +∞ n −x = x e dx, (1) Science 0 of

(2n)! √ 1 2n π = Γ n +

2 n! 2 Journal Z +∞ e−x = xn √ dx. (2) x Global 0 The Betâ function is given in terms of the integral Z 1 B(s, t) = xs−1(1 − x)t−1 dx, 0. 0

Author: Faculté des Sciences de Gabès, Département de Mathématiques, Cité Erriadh, 6072 Gabès, Tunisie. e-mail: [email protected]

which is symmetric in s and t, i.e., B(s, t) = B(t, s). Notice that, after a change of variable 1 x = 1+y , we get Z +∞ xs−1 B(s, t) = s+t dx. 0 (1 + x) This function also admits the following representation in terms of the Gamma function [3] Γ(s)Γ(t) B(s, t) = . Γ(s + t) Ref In particular, if s and t are non-zero integers, then we have

201 n!p!

= B(n + 1, p + 1) 3. (n + p + 1)! ear P. de Y Z 1

n p Maroni, 2 = x (1 − x) dx, (3) l'Ing 0 +∞ n Z x é Trait nieur, = n+p+2 dx, n, p ≥ 0. (4)

0 (1 + x)

F V onc The monic Hermite polynomials Hn(x) are orthogonal in the interval (−∞, +∞) with V 2

respect to the weight function e−x and fulﬁl the following orthogonality relation [2] Eul tions ue ersion I s s √ é Z +∞ I 2 π −x G e Hn(x)Hm(x) dx = n!δn,m, n, m ≥ 0, m é XV

−∞ 2 n

é é ralit riennes where δn,m is the Kronecker delta. The canonical moments, (H) , of the Hermite polynomials have the representation [3] n≥0 é s (Sciences Fondamentales) A 154 Paris, 1994. 1 )

n ,

F 1 + (−1) n!

) = (H) P n+1 n n o

2 Γ( 2 + 1) lyn

Z +∞ ô

= xne−x2 dx, n ≥ 0. (5) m

Research Volume e −∞ Orthogonauxs Classiques b) Some other integrals

Frontier The Wallis integral is given by

π Z 2 n In = sin x dx, n ≥ 0. Science 0 of By a simple integration by parts, we can obtain (2n)!π 22n(n!)2 Journal I = ,I = , n ≥ 0. 2n 22n+1(n!)2 2n+1 (2n + 1)! . Te

Global By the change of variable t = sin x, this gives the following formulas c hnique (2n)!π Z 1 x2n

= √ dx, n ≥ 0, (6) -30. 2n+1 2 2 2 (n!) 0 1 − x s 22n(n!)2 Z 1 x2n+1 = √ dx, n ≥ 0. (7) 2 (2n + 1)! 0 1 − x Now, let consider the following integral

Certain Sequences and its Integral Representations in Terms of Laguerre Polynomials

π Z 4 n Tn = tan x dx, n ≥ 0. 0 It is easy to see that 1 T + T = , n ≥ 0. n+2 n n + 1 We get by iteration the two following formulas

n n+k X (−1) n otes T2n+1 = + (−1) T1, n ≥ 0. N 2k k=1 n n+k X (−1) n+1 201 T2n+2 = + (−1) T0, n ≥ 0. 2k + 1 r k=0 ea Y

Then, by the change of variable t = tan x, we get 3 n (−1)n X (−1)k Z 1 x2n+1 ln 2 + = dx, n ≥ 0, (8) 2 k 1 + x2 k=1 0 V n−1 π X (−1)k+1 Z 1 x2n V (−1)n + = dx, n ≥ 0, (9) 2 ue ersion I 4 2k + 1 1 + x s k=0 0 s I

P0 P−1 XV with the convention k=1 = k=0 = 0. We also consider the following integral

Z 1 n

x )

Rn = dx, n ≥ 0. F ) 0 1 + x

1 It is easy to see that Rn + Rn+1 = n , n ≥ 1, and hence the following formula

n Research Volume X (−1)k Z 1 xn (−1)n ln 2 + = dx, n ≥ 0. (10) k 1 + x k=1 0 Frontier Finally, we consider the integral 1 Z 1 B = (1 − x)nex dx, n ≥ 0. Science n n! 0 of

1 For n ≥ 1, integration by parts yields Bn = Bn−1 − n! , and we obtain the formula

n Journal X 1 1 Z 1 e − = (1 − x)nex dx, n ≥ 0. k! n! 0

k=0 Global This gives, after a change of variable t = 1 − x, the following relation

n 1 X 1 Z 1 n! 1 − = xne−x dx, n ≥ 0. (11) e k! k=0 0 In this paper, we introduce the following connection formula, between the monomial n {x }n≥0 and the shifted Laguerre polynomials,

Certain Sequences and its Integral Representations in Terms of Laguerre Polynomials

Z +∞ n 1 m −t (m) x = t e Ln t(x + 1) dt, n ≥ 0, m ∈ N \{0}. (n + m)! 0 As an application of our formula, we give the integral representations in terms of Laguerre polynomials for the sequences given by the equations (1)−(11). II. Integral Representations in Terms of Laguerre Polynomials (m) Let {Ln }n≥0 be the monic Laguerre polynomial sequence, with parameter m ∈ N \{0}, [4] Ref n X n (n + m)! L(m)(x) = (−1)n−ν xν, n ≥ 0. (12) 201 n ν (ν + m)! ν=0 1. ear

Y For any c ∈ and any polynomial p, let introduce in the space of polynomials the linear York, 1971. C H.

4 isomorphism Sc, called intertwining operator, and given by [1]: Hoc

+∞ Z hstadt, −t Sc(p)(x) = te p t(x − c) + c dt. 0 V

V The operator Sc can be characterized taking into account its linearity as well as the fact T he ue ersion I

s Functions of Mathematica s n n

I Sc (x − c) = (n + 1)!(x − c) , n ≥ 0. (13) XV By (13) and (12), it is easy to prove that

m−1 (m) m−1 n S0 x Ln (x) = (n + m)!x (x − 1) , n ≥ 0. )

F ) Hence, we can obtain the following result. Theorem 2.1 For every integer m ∈ N \{0}, the following formula holds Z +∞ Research Volume n 1 m −t (m) x = t e Ln t(x + 1) dt, n ≥ 0. (14)

(n + m)! 0 l Physics

Frontier Now, as an application of the above formula, we can express the sequences given by the equations (1)−(11) by integral representations in terms of Laguerre polynomials. Indeed, .

substituting expression (14) into (1)−(11), we can state the following theorem. D ov Science er Publications Inc. New

of Theorem 2.2 For every integers m ≥ 1, and n, p ≥ 0, the following formulas hold

Z +∞ Z +∞ n!(n + m)! = tme−(x+t)L(m)t(x + 1) dtdx Journal n 0 0 √ (2n)!m! π n + m Z +∞ Z +∞ tm Global √ −(x+t) (m) n = e Ln t(x + 1) dtdx 4 n 0 0 x

Z 1 Z +∞ n!p!(n + m)! p m −t (m) = (1 − x) t e Ln t(x + 1) dtdx (n + p + 1)! 0 0

Z +∞ Z +∞ m −t n!p!(n + m)! t e (m) = n+p+2 Ln t(x + 1) dtdx (n + p + 1)! 0 0 (1 + x)

Certain Sequences and its Integral Representations in Terms of Laguerre Polynomials

n +∞ +∞ 1 + (−1) n!(n + m)! Z Z 2 = tme−(x +t)L(m)t(x + 1) dtdx n+1 n n 2 Γ + 1 −∞ 0 2 (2n)!(2n + m)!π Z 1 Z +∞ tme−t = √ L(m)t(x + 1) dtdx 2 2n+1 2 2n (n!) 2 0 0 1 − x

(n!)2(2n + m + 1)!22n Z 1 Z +∞ tme−t Notes = √ L(m) t(x + 1) dtdx 2 2n+1 (2n + 1)! 0 0 1 − x

n n k Z 1 Z +∞ m −t 201 (2n + m + 1)!(−1) X (−1) t e (m)

r ln 2 + = L2n+1 t(x + 1) dtdx 2 k 1 + x2 ea k=1 0 0 Y 5 n−1 k+1 Z 1 Z +∞ m −t π X (−1) t e (m) (2n + m)!(−1)n + = L t(x + 1) dtdx 4 2k + 1 1 + x2 2n k=0 0 0

n (−1)k Z 1 Z +∞ tme−t V

X V (n + m)!(−1)n ln 2 + = L(m)t(x + 1) dtdx k 1 + x n 0 0 ue ersion I

k=1 s

s I n Z 1 Z +∞ 1 X 1 XV n!(n + m)! 1 − = tme−(t+x)L(m)t(x + 1) dtdx e k! n k=0 0 0

with the convention P0 = P−1 = 0.

k=1 k=0 ) F ) Corollary 2.3 For m = 1 and p = 0 we have, for every integer n ≥ 0, the following special cases

Z +∞ Z +∞ −(x+t) (1) Research Volume n!(n + 1)! = te Ln t(x + 1) dtdx 0 0 √ (2n)!(n + 1) π Z +∞ Z +∞ t Frontier √ −(x+t) (1) 2n = e Ln t(x + 1) dtdx 2 0 0 x Science Z 1 Z +∞ −t (1) of n! = te Ln t(x + 1) dtdx 0 0 Journal Z +∞ Z +∞ −t te (1) n! = n+2 Ln t(x + 1) dtdx 0 0 (1 + x) Global

n +∞ +∞ 1 + (−1) n!(n + 1)! Z Z 2 = te−(x +t)L(1)t(x + 1) dtdx n+1 n n 2 Γ 2 + 1 −∞ 0

(2n + 1)!π 2n Z 1 Z +∞ te−t = √ L(1)t(x + 1) dtdx 2n+1 2 2n 2 n 0 0 1 − x

Certain Sequences and its Integral Representations in Terms of Laguerre Polynomials

Z 1 Z +∞ te−t n!(n + 1)22n+1 = √ L(1) t(x + 1) dtdx 2 2n+1 0 0 1 − x

n n k Z 1 Z +∞ −t (2n + 2)!(−1) X (−1) te (1) ln 2 + = L t(x + 1) dtdx 2 k 1 + x2 2n+1 k=1 0 0

n−1 k+1 Z 1 Z +∞ −t π X (−1) te (1) (2n + 1)!(−1)n + = L t(x + 1) dtdx 4 2k + 1 1 + x2 2n Notes k=0 0 0 n k Z 1 Z +∞ −t 201 X (−1) te (n + 1)!(−1)n ln 2 + = L(1)t(x + 1) dtdx k 1 + x n ear k=1 0 0 Y

n 1 +∞ 6 1 X 1 Z Z n!(n + 1)! 1 − = te−(t+x)L(1)t(x + 1) dtdx e k! n k=0 0 0 P0 P−1 V with the convention k=1 = k=0 = 0. V ue ersion I

s Remark 2.1 Note that, if we take n even in the fifth formula, we obtain

s I +∞ +∞ (2n)!(2n + 1)! Z Z 2 XV −(x +t) (1) 2n = te L2n t(x + 1) dtdx. 2 n! −∞ 0

III. Acknowledgements )

F ) Sincere thanks are due to the referee for his/her careful reading of the manuscript and for his/her valuable comments.

References Références Referencias Research Volume 1. H. Hochstadt, The Functions of Mathematical Physics. Dover Publications Inc. New York, 1971.

Frontier 2. N. N. Lebedev, Special Functions and their Applications. Revised English Edition, Dover Publications, New York, 1972. 3. P. Maroni, Fonctions Eulériennes, Polynômes Orthogonaux Classiques. Techniques Science de l'Ingénieur, Traité Généralités (Sciences Fondamentales) A 154 Paris, 1994. 1-30. of 4. G. Szeg¨o, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, Amer. Math. Soc., Providence, RI, 1975. Journal Global