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Mahler Polynomial
In mathematics, the Mahler polynomials ''g''''n''(''x'') are polynomials introduced by in his work on the zeros of the incomplete gamma function. Mahler polynomials are given by the generating function :\displaystyle \sum g_n(x)t^n/n! = \exp(x(1+t-e^t)) Mahler polynomials can be given as the Sheffer sequence In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are ... for the functional inverse of 1+''t''–''e''''t'' . The first few examples are :g_0=1; :g_1=0; :g_2=-x; :g_3=-x; :g_4=-x+3x^2; :g_5=-x+10x^2; :g_6=-x+25x^2-15x^3; :g_7=-x+56x^2-105x^3; :g_8=-x+119x^2-490x^3+105x^4; References * *{{Citation , last1=Roman , first1=Steven , title=The umbral calculus , url=https://books.google.com/books?id=JpHjkhFLfpgC , publisher=Academic Press Inc. arcourt Brace Jovanovich Publishers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incomplete Gamma Function
In mathematics, the upper and lower incomplete gamma functions 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 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. Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity. Definition The upper incomplete gamma function is defined as: \Gamma(s,x) = \int_x^ t^\,e^\, dt , whereas the lower incomplete gamma function is defined as: \gamma(s,x) = \int_0^x t^\,e^\, dt . In both cases is a complex parameter, such that the real part of is positive. Properties By integration by parts we find the recurrence relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheffer Sequence
In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are named for Isador M. Sheffer. Definition Fix a polynomial sequence (''p''''n''). Define a linear operator ''Q'' on polynomials in ''x'' by :Qp_n(x) = np_(x)\, . This determines ''Q'' on all polynomials. The polynomial sequence ''p''''n'' is a ''Sheffer sequence'' if the linear operator ''Q'' just defined is ''shift-equivariant''; such a ''Q'' is then a delta operator. Here, we define a linear operator ''Q'' on polynomials to be ''shift-equivariant'' if, whenever ''f''(''x'') = ''g''(''x'' + ''a'') = ''T''''a'' ''g''(''x'') is a "shift" of ''g''(''x''), then (''Qf'')(''x'') = (''Qg'')(''x'' + ''a''); i.e., ''Q'' commutes with every shift operator: ''T''''a''''Q'' = ''QT''''a''. Properties The set of all Sheffer sequences is a group un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
