## On Laguerres Series Third Note

In this paper, we propose a fractional generalization of the well-known Laguerre differential equation. We then apply the Frobenius method with the fractional power series expansion to obtain two linearly independent solutions of the problem. For certain eigenvalues, the infinite series solution truncate to obtain the singular and non-singular fractional Laguerre functions.

We obtain the fractional Laguerre functions in closed forms, and establish their orthogonality result.

## numpy/comdaisismaper.gq at master · numpy/numpy · GitHub

The applicability of the new fractional Laguerre functions is illustrated. In recent years, there are interests in studying fractional Sturm-Liouville eigenvalue problems. For instance, the fractional Bessel equation with applications was investigated in Okrasinski and Plociniczak [ 1 , 2 ], where the fractional derivative is of the Riemann-Liouville type.

In Abu Hammad and Khalil [ 3 ] the authors solved the fractional Legendre equation with conformable derivative and established the orthogonality property of the fractional Legendre functions. The applications of the fractional Legendre functions in solving fractional differential equations, were illustrated in Kazema et al. In this project we propose the following fractional generalization of the well-known Laguerre differential equation. The conformable derivative was introduced recently in Khalil et al.

Definition 1. The conformable derivative is a local derivative which has a physical and a geometrical interpretations and potential applications in physics and engineering [ 7 , 8 ]. It satisfies the nice properties of the integer derivative such as, the product rule, the quotient rule, and the chain rule, and it holds that. For more details about the conformable derivative we refer the reader to Abdeljawad [ 9 ] and Khalil et al.

We mention here that even though the conformable is a nonlocal derivative see [ 10 , 11 ] , the simplicity and applications of the derivative make it of interests. Also, the applications of the obtained Fractional Leguerre functions are indicated in this manuscript. The rest of the paper is organized as follows: In section 2, we apply the Frobenius method together with the fractional series solution to solve the above equation and to obtain the fractional Laguerre functions. In section 3, we establish the orthogonality result of the fractional Laguerre functions and present the fractional Laguerre functions for several eigenvalues.

Finally, we close up with some concluding remarks in section 4.

## On Laguerre's Series: Third Note.

The series solution is commonly used to solve various types of fractional differential equations see [ 12 — 16 ]. Lemma 2. The coefficients a n in Equation 2. Proof: The proof can be easily obtained by iterating the recursion in 2. Remark 2. Now, in Equation 2. Analogously, in Equation 2. Theorem 3. Thus, the equation is of a special type of the fractional Sturm-liouville eigenvalue problem. Thus the right hand side of Equation 3.

Remark 3.

Since the fractional Laguerre functions are orthogonal, they can be used as a basis of the spectral method to study fractional differential equations analytically and numerically. They also can be used as a basis of the fractional Gauss-Laguerre quadrature for approximating the value of integrals of the form.

On generating functions of biorthogonal polynomials suggested by the Laguerre polynomials. Bengal Engineering and Science University. In this note, we have obtained some novel bilateral generating functions involving Konhauser biorthogonal polynomials, which is converted into trilateral generating functions with Tchebycheff polynomials by group theoretic method. As special cases, we have obtained the corresponding results on generalised Laguerre polynomials.

Some applications are also given here. Keywords : Laguerre polynomials,biorthogonal polynomials, generating functions. Standards 46, pp. New York: McGraw-Hill, There are different cases obtained, depending on the kind of the s x function in Eq. When s x is a constant, Eq. When s x is a polynomial of the first degree, Eq.

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There are three different cases when s x is a polynomial of the second degree. When the second degree polynomial has two different real roots, Eq. When the second degree polynomial has one double root, Eq. Finally, when the second degree polynomial has two complex roots, Eq. These results are summarized in Table 1. Polynomials obtained depending on the s x function of Eq.

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## A note on the use of Laguerre polynomials in the inversion of Laplace transforms

The Sturm-Liouville Theory is covered in most advanced physics and engineering courses. The analysis of this equation and its solutions is called the Sturm-Liouville theory. There, the close analogy of this theory with linear algebra concepts is also shown. For example, functions here take the role of vectors there, and linear operators here take that of matrices there. The next section shows some of the most important applications of Hermite, Gegenbauer, Tchebycheff, Laguerre and Legendre polynomials in applied Mathematics and Physics.

These polynomials are of great importance in mathematical physics, the theory of approximation, the theory of mechanical quadrature, engineering, and so forth.

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Laguerre polynomials were named after Edmond Laguerre — Laguerre studied a special case in , and in , Nikolay Yakovlevich Sonin worked on the general case known as Sonine polynomials, but they were anticipated by Robert Murphy The Laguerre differential equation and its solutions, that is, Laguerre polynomials, are found in many important physical problems, such as in the description of the transversal profile of Laguerre-Gaussian laser beams [ 4 ].

This equation is. The radial part, R r , satisfies the equation. Hermite polynomials were defined into the theory of probability by Pierre-Simon Laplace in , and Charles Hermite extended them to include several variables and named them in [ 7 ]. Hermite polynomials are used to describe the transversal profile of Hermite-Gaussian laser beams [ 4 ], but mainly to analyze the quantum mechanical simple harmonic oscillator [ 8 ]. The oscillating particle has mass m and total energy E. By use of the abbreviations. Hermite polynomials also appear in probability as the Edgeworth series, in combinatorics as an example of an Appell sequence, obeying the umbral calculus, in numerical analysis as Gaussian quadrature, etc.

Legendre polynomials were first introduced in by Adrien-Marie Legendre.

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Spherical harmonics are an important class of special functions that are closely related to these polynomials. Normalizing the associated Legendre function, one obtains the orthonormal function. Taking the product of Eqs. Legendre polynomials are frequently encountered in physics and other technical fields. Some examples are the coefficients in the expansion of the Newtonian potential that gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge, the gravitational and electrostatic potential inside a spherical shell, steady-state heat conduction problems in spherical problems inside a homogeneous solid sphere, and so forth [ 11 ].

Tchebycheff polynomials, named after Pafnuty Tchebycheff also written as Chebyshev, Tchebyshev or Tschebyschow , are important in approximation theory because the roots of the Tchebycheff polynomials of the first kind, which are also called Tchebycheff nodes, are used as nodes in polynomial interpolation.

Approximation theory is concerned with how functions can best be approximated with simpler functions, and through quantitatively characterizing the errors introduced thereby.