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Laguerre Polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are solutions of Laguerre's equation: x y ″ + ( 1 − x ) y ′ + n y = 0 displaystyle xy''+(1-x)y'+ny=0 which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. More generally, the name Laguerre polynomials
Laguerre polynomials
is used for solutions of x y ″ + ( α + 1 − x ) y ′ + n y = 0   . displaystyle xy''+(alpha +1-x)y'+ny=0~
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Mathematics
Mathematics
Mathematics
(from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5] It has no generally accepted definition.[6][7] Mathematicians seek out patterns[8][9] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist
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Multiplication Theorem
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name. The various relations all stem from the same underlying principle; that is, the relation for one special function can be derived from that for the others, and is simply a manifestation of the same identity in different guises.Contents1 Finite characteristic 2 Gamma function–Legendre formula 3 Polygamma function, harmonic numbers 4 Hurwitz zeta function 5 Periodic zeta function 6 Polylogarithm 7 Kummer's function 8 Bernoulli polynomials 9 Bernoulli map 10 Characteristic zero 11 Notes 12 ReferencesFinite characteristic[edit] The multiplication theorem takes two common forms. In the first case, a finite number of terms are added or multiplied to give the relation. In the second case, an infinite number of terms are added or multiplied
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Christoffel–Darboux Formula
In mathematics, the Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin Bruno Christoffel (1858) and Jean Gaston Darboux (1878)
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Turán's Inequalities
In mathematics, Turán's inequalities are some inequalities for Legendre polynomials
Legendre polynomials
found by Paul Turán (1950) (and first published by Szegö (1948)). There are many generalizations to other polynomials, often called Turán's inequalities, given by (E. F. Beckenbach, W. Seidel & Otto Szász 1951) and other authors. If Pn is the nth Legendre polynomial, Turán's inequalities state that P n ( x ) 2 > P n − 1 ( x ) P n + 1 ( x )  for  − 1 < x < 1. displaystyle ,!P_ n (x)^ 2 >P_ n-1 (x)P_ n+1 (x) text for -1<x<1
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Hilbert Space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space
Hilbert space
is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz
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Lp Space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz
Frigyes Riesz
(Riesz 1910). Lp spaces form an important class of Banach
Banach
spaces in functional analysis, and of topological vector spaces
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If And Only If
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false). It is controversial whether the connective thus defined is properly rendered by the English "if and only if", with its pre-existing meaning
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Monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:(1): A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. The constant 1 is a monomial, being equal to the empty product and x0 for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power xn of x, with n a positive integer
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Incomplete Gamma Function
In mathematics, the upper incomplete gamma function and lower incomplete gamma function are types of special functions, which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which are defined similarly to the gamma function, another type of special function, but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit
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Arthur Erdélyi
Arthur Erdélyi FRS,[1] FRSE
FRSE
(2 October 1908 – 12 December 1977) was a Hungarian-born British mathematician. Erdélyi was a leading expert on special functions, particularly orthogonal polynomials and hypergeometric functions.[2][3]Contents1 Biography 2 Family 3 Research 4 Works 5 Awards 6 ReferencesBiography[edit] He was born Arthur Diamant in Budapest, Hungary
Hungary
to Ignác Josef Armin Diamant and Frederike Roth. His name was changed to Erdélyi when his mother remarried to Paul Erdélyi. He attended the primary and secondary schools there from 1914 to 1926. His interest in mathematics dates back to this time. Erdélyi was a Jew, and so it was difficult for him to receive a university education in his native Hungary. He travelled to Brno, Czechoslovakia, to obtain a degree in electrical engineering
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Hypergeometric Function
In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Erdélyi et al. (1953) and Olde Daalhuis (2010). There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities
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Contour Integral
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.[1][2][3] Contour integration
Contour integration
is closely related to the calculus of residues,[4] a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods.[5]
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Pochhammer Symbol
In mathematics, the falling factorial (sometimes called the descending factorial,[1] falling sequential product, or lower factorial) is defined as ( x ) n = x n _ = x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) = ∏ k = 1 n ( x − ( k − 1 ) ) = ∏ k = 0 n − 1 ( x − k ) . displaystyle (x)_ n =x^ underline n =x(x-1)(x-2)cdots (x-n+1)=prod _ k=1 ^ n (x-(k-1))=prod _ k=0 ^ n-1 (x-k)
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Mehler Kernel
Contents1 Mehler's formula 2 Physics version 3 Probability version 4 Fractional Fourier transform 5 See also 6 ReferencesMehler's formula[edit] Mehler (1866) defined a function[1] E ( x , y ) = 1 1 − ρ 2 exp ⁡ ( − ρ 2 ( x 2 + y 2 ) − 2 ρ x y ( 1 − ρ 2


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Math. Ann.
Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin.[1] Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy.[2] Between 1945 and 1947 the journal briefly ceased publication.[1] References[edit]^ a b Behnke, Heinrich (March 1973). "Rückblick auf die Geschichte der Mathematischen Annalen". Math
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