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 picture info 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~ [...More...] "Laguerre Polynomials" on: Wikipedia Google Yahoo Parouse picture info 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 [...More...] "Mathematics" on: Wikipedia Google Yahoo Parouse 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 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 [...More...] "Multiplication Theorem" on: Wikipedia Google Yahoo Parouse 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) [...More...] "Christoffel–Darboux Formula" on: Wikipedia Google Yahoo Parouse 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
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