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Some indefinite integrals involving certain polynomial.

[section]1. Introduction and preliminaries

Definition 1.1. Gegenbauer polynomials or ultraspherical polynomials [C.sup.[alpha].sub.n](x) are orthogonal polynomials on the interval [-1,1] with respect to the weight function [(1-[x.sup.2]).sup.[alpha]-1/2]. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named for Leopold Gegenbauer.

Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation

(1-[x.sup.2])y"-(2[alpha] + 1)xy' + n(n + 2[alpha])y = 0.

When [alpha] = 1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the Legendre polynomials.

They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite

[C.sup.[alpha].sub.n](z) = [[[(2[alpha]).sub.n]/n!].sub.2][F.sub.1](-n, (2[alpha] + n);[alpha] + 1/2;[1-z/2]).

They are special cases of the Jacobi polynomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

One therefore also has the Rodrigues formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 1.2. The Pochhammer's symbol is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where b is neither zero nor negative integer and the notation [GAMMA] stands for gamma function.

Definition 1.3. In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the 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 crossings 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.

The Bernoulli polynomials [B.sub.n](x) admit a variety of different representations. Which among them should be taken to be the definition may depend on one's purposes.

Definition 1.4. Explicit formula is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for n [greater than or equal to] 0, where [B.sub.k] are the Bernoulli numbers.

Definition 1.5. In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator. They are also used in systems theory in connection with nonlinear operations on Gaussian noise. They are named after Charles Hermite (1864) although they were studied earlier by Laplace (1810) and Chebyshev (1859).

There are two different standard ways of normalizing Hermite polynomials:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(the "probabillsts' Hermite polynomials"), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(the "physicists' Hermite polynomials").

These two definitions are not exactly equivalent; either is a rescaling of the other, to wit

[H.sub.n](x) = [2.sup.n/2][He.sub.n]([square root of 2x]), [He.sub.n](x) = [2.sup.-n/2][H.sub.n](x/[square root of 2]).

The notation He and H is that used in the standard references Tom H. Koornwinder, Roderick S. C. Wong, and Roelof Koekoek et al(2010), and Abramowitz & Stegun. The polynomials [He.sub.n] are sometimes denoted by [H.sub.n], especially in probability theory, because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the probability density function for the normal distribution with expected value 0 and standard deviation 1.

The first eleven probabiists' Hermite polynomials are:

[He.sub.0](x) = 1, [He.sub.1](x) = x, [He.sub.2](x) = [x.sup.2] - 1, [He.sub.3](x) = [x.sup.3] - 3x, [He.sub.4](x) = [x.sup.4] - 6[x.sup.2] + 3, [He.sub.5](x) = [x.sup.5] - 10[x.sup.3] + 15x, [He.sub.6](x) = [x.sup.6] - 15[x.sup.4] + 45[x.sup.2] - 15, [He.sub.7](x) = [x.sup.7] - 21[x.sup.5] + 105[x.sup.3] - 105x, [He.sub.8](x) = [x.sup.8] - 28[x.sup.6] + 210[x.sup.4] - 420[x.sup.2] + 105, [He.sub.9](x) = [x.sup.9] - 36[x.sup.7] + 378[x.sup.5] - 1260[x.sup.3] + 945x, [He.sub.10](x) = [x.sup.10] - 45[x.sup.8] + 630[x.sup.6] - 3150[x.sup.4] + 4725[x.sup.2] - 945,

and the first eleven physicists' Hermite polynomials are:

[H.sub.0](x) = 1, [H.sub.1](x) = 2x, [H.sub.2](x) = 4[x.sup.2] - 2, [H.sub.3](x) = 8[x.sup.3] - 12x, [H.sub.4](x) = 16[x.sup.4] - 48[x.sup.2] + 12, [H.sub.5](x) = 32[x.sup.5] - 160[x.sup.3] + 120x, [H.sub.6](x) = 64[x.sup.6] - 480[x.sup.4] + 720[x.sup.2] - 120, [H.sub.7](x) = 128[x.sup.7] - 1344[x.sup.5] + 3360[x.sup.3] - 1680x, [H.sub.8](x) = 256[x.sup.8] - 3584[x.sup.6] + 13440[x.sup.4] - 13440[x.sup.2] + 1680, [H.sub.9](x) = 512[x.sup.9] - 9216[x.sup.7] + 48384[x.sup.5] - 80640[x.sup.3] + 30240x, [H.sub.10](x) = 1024[x.sup.10] - 23040[x.sup.8] + 1612S0[x.sup.6] -403200[x.sup.4] + 302400[x.sup.2] - 30240.

Definition 1.6. The sequence of Lucas polynomials is a sequence of polynomials defined by the recurrence relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first few Lucas polynomials are:

[L.sub.0](x) = 2,

[L.sub.1](x) = x,

[L.sub.2](x) = [x.sup.2] + 2,

[L.sub.3](x) = [x.sup.3] + 3x,

[L.sub.4](x) = [x.sup.4] + 4[x.sup.2] + 2.

Definition 1.7. In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers.

These Fibonacci polynomials are defined by a recurrence relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first few Fibonacci polynomials are:

[F.sub.0](x) = 0, [F.sub.1](x) = 1, [F.sub.2](x) = x, [F.sub.3](x) = [x.sup.2] + 1, [F.sub.4](x) = [x.sup.3] + 2x.

Definition 1.8. The polylogarithm (also known as Jonquie's function) is a special function [Li.sub.s](z) that is defined by the infinite sum, or power series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

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 [absolute value of z] < 1.

Definition 1.9. Generalized ordinary hypergeometric function of one variable is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

or

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where denominator parameters [b.sub.1], [b.sub.2], ..., [b.sub.B] are neither zero nor negative integers and A, B are non-negative integers.

[section]2. Main indefinite integrals

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[section]3. Derivation of the integrals

Involving the method of same type of [6], one can derive the integrals.

[section]4. Conclusion

In our work we have established certain indefinite integrals involving Fibonacci polynomials, Lucas polynomials, Bernoulli polynomials and Hermite polynomials. We hope that the development presented in this work will stimulate further interest and research in this important area of Mathematics.

References

[1] Abramowitz, A. Milton and Irene Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards, 1970.

[2] Frank Olver, W. J. Lozier, M. Daniel, Boisvert, F. Ronald and C. W. Clark. NIST, Handbook of Mathematical Functions, Cambridge University Press, 2010.

[3] G. Arfken, Mathematical Methods for Physicists, Academic Press, 1985.

[4] Nico Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, New York, 1996.

[5] Norbert Wiener, The Fourier Integral and Certain of its Applications, New York, Dover Publications, 1958.

[6] Ricci and Paolo Emilio, Generalized Lucas polynomials and Fibonacci polynomials, Rivista di Matematica della Universit di Parma, 4(1995), 137-146.

[7] Salahuddin, Hypergeometric Form of Certain Indefinite Integrals, Global Journal of Science Frontier Research(F), 6(2012), No. 12, 37-41.

[8] Zhiwei Sun and Hao Pan, Identities concerning Bernoulli and Euler polynomials, Acta Arithmetica, 125(2006), 21-39.

Salahuddin

Mewar University, Gangrar, Chittorgarh (Rajasthan), India

E-mail: vsludn@gmail.com, sludn@yahoo.com
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Author:Salahuddin
Publication:Scientia Magna
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Date:Sep 1, 2013
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