the Legendre rational functions are a sequence of orthogonal functions on [0, ∞). They are obtained by composing the Cayley transform with Legendre polynomials...

related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials...

procedure results in families of rational orthogonal functions called Legendre rational functions and Chebyshev rational functions. Solutions of linear differential...

function Riesz function Hypergeometric functions: Versatile family of power series. Confluent hypergeometric function Associated Legendre functions Meijer G-function...

"Legendre's Chi Function". MathWorld. Djurdje Cvijović, Jacek Klinowski (1999). "Values of the Legendre chi and Hurwitz zeta functions at rational arguments"...

Cornelius Lanczos, for example, called it "void of any rationality" and would instead use z!). Legendre's normalization does simplify some formulae, but complicates...

reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed...

elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they...

filter Legendre form Legendre function Legendre moment Legendre polynomials Legendre pseudospectral method Legendre rational functions Legendre relation...

Klinowski, Djurdje Cvijović (1999). "Values of the Legendre chi and Hurwitz zeta functions at rational arguments". Mathematics of Computation. 68 (228):...

can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms, also known as the elliptic integrals...

(z_{1}),\operatorname {Re} (z_{2})>0} . The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its...

Ramanujan tau function All Dirichlet characters are completely multiplicative functions, for example ( n / p ) {\displaystyle (n/p)} , the Legendre symbol,...

transform is used to adapt Legendre polynomials for use with functions on the positive real numbers with Legendre rational functions. As a real homography...

between the vectors x and x1. The functions P i : [ − 1 , 1 ] → R {\displaystyle P_{i}:[-1,1]\to \mathbb {R} } are the Legendre polynomials, and they can be...

Floor and ceiling functions In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer...

integrable functions on the unit sphere S2 is a Hilbert space. The basis functions in this case are the spherical harmonics. The Legendre polynomials...

functions. These include most of the commonly used functions of mathematical physics. Legendre functions are solutions of a second order differential equation...

flawed proof that π cannot be rational; Legendre (1794) completed the proof and showed that π is not the square root of a rational number. Liouville (1840)...

hypergeometric functions what is now called Gauss's continued fractions. They can be used to express many elementary functions and some more advanced functions (such...

théorie des nombres (1798), Adrien-Marie Legendre derives a necessary and sufficient condition for a rational number to be a convergent of the continued...

number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative...

coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined...

growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately x log ⁡ x {\displaystyle...

In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied...

been extended over the centuries to include zero (0), negative numbers, rational numbers such as one half ( 1 2 ) {\displaystyle \left({\tfrac {1}{2}}\right)}...

reciprocity—Let p and q be distinct odd prime numbers, and define the Legendre symbol as ( q p ) = { 1 if  n 2 ≡ q mod p  for some integer  n − 1 otherwise...

Leonhard Euler (1738), Kausler (1802), Peter Barlow (1811), Adrien-Marie Legendre (1830), Schopis (1825), Olry Terquem (1846), Joseph Bertrand (1851), Victor...

coefficient generating functions arising from the sequence factorial function multiplier provided by the rational convergent functions include n ! 2 = [ z...

prime-counting functions. This article provides links to functions of both classes. An example of an arithmetic function is the divisor function whose value...