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Mittag-Leffler Polynomials
In mathematics, the Mittag-Leffler polynomials are the polynomials ''g''''n''(''x'') or ''M''''n''(''x'') studied by . ''M''''n''(''x'') is a special case of the Meixner polynomial ''M''''n''(''x;b,c'') at ''b = 0, c = -1''. Definition and examples Generating functions The Mittag-Leffler polynomials are defined respectively by the generating functions : \displaystyle \sum_^ g_n(x)t^n :=\frac\Bigl(\frac \Bigr)^x and : \displaystyle \sum_^ M_n(x)\frac:=\Bigl(\frac \Bigr)^x=(1+t)^x(1-t)^=\exp(2x\text t). They also have the bivariate generating function : \displaystyle \sum_^\sum_^ g_n(m)x^my^n =\frac. Examples The first few polynomials are given in the following table. The coefficients of the numerators of the g_n(x) can be found in the OEIS, though without any references, and the coefficients of the M_n(x) are in the OEIS as well. : Properties The polynomials are related by M_n(x)=2\cdot \, g_n(x) and we have g_n(1)=1 for n\geqslant 1 . Also g_(\frac12)=g_(\frac12)=\frac12\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meixner Polynomial
In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by . They are given in terms of binomial coefficients and the (rising) Pochhammer symbol In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ... by :M_n(x,\beta,\gamma) = \sum_^n (-1)^kk!(x+\beta)_\gamma^ See also * Kravchuk polynomials References * * * * * * * * * * * * * *{{cite journal , first1= Xiang-Sheng , last1=Wang , first2=Roderick , last2=Wong , title= Global asymptotics of the Meixner polynomials , journal = Asymptot. Anal. , year=2011 , volume=75 , number=3–4 , pages=211–231 , doi=10.3233/ASY-2011-1060 , arxiv=1101.4370 Orthogonal polynomials ... [...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] |
Binomial Type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers \left\ in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities :p_n(x+y)=\sum_^n\, p_k(x)\, p_(y). Many such sequences exist. The set of all such sequences forms a Lie group under the operation of umbral composition, explained below. Every sequence of binomial type may be expressed in terms of the Bell polynomials. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type). Polynomial sequences put on firm footing the vague 19th century notions of umbral calculus. Examples * In consequence of this definition the binomial theorem can be stated by saying that the sequence is of binomial type. * The sequence of " lower factorials" is defined by(x)_n=x(x-1)(x-2)\cdot\cdots\cdot(x-n+1).(In the theory of special functions, this same notation denotes upper fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The National Academy Of Sciences
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Sciences, published since 1915, and publishes original research, scientific reviews, commentaries, and letters. According to ''Journal Citation Reports'', the journal has a 2021 impact factor of 12.779. ''PNAS'' is the second most cited scientific journal, with more than 1.9 million cumulative citations from 2008 to 2018. In the mass media, ''PNAS'' has been described variously as "prestigious", "sedate", "renowned" and "high impact". ''PNAS'' is a delayed open access journal, with an embargo period of six months that can be bypassed for an author fee ( hybrid open access). Since September 2017, open access articles are published under a Creative Commons license. Since January 2019, ''PNAS'' has been online-only, although print issues are a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Eta Function
In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: \eta(s) = \sum_^ = \frac - \frac + \frac - \frac + \cdots\approx \prod_^ \infty . This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ''ζ''(''s'') — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ''ζ''*(''s''). The following relation holds: \eta(s) = \left(1-2^\right) \zeta(s) Both Dirichlet eta function and Riemann zeta function are special cases of Polylogarithm. While the Dirichlet series expansion for the eta function is convergent only for any complex number ''s'' with real part > 0, it is Abel summable for any complex number. This serves to define the eta function as an entire function. (The above relation and the facts that the eta function is ent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula \frac where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f''; that is, the infinitesimal absolute change in ''f,'' namely f', scaled by the current value of ''f.'' When ''f'' is a function ''f''(''x'') of a real variable ''x'', and takes real, strictly positive values, this is equal to the derivative of ln(''f''), or the natural logarithm of ''f''. This follows directly from the chain rule: \frac\ln f(x) = \frac \frac Basic properties Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does ''not'' take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have (\log uv)' = (\log u + \log v)' = (\log u)' + (\log v)' . So for positive-real-valued functions, the logarithmic deri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > 1 and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Reflection Formula
In mathematics, a reflection formula or reflection relation for a function ''f'' is a relationship between ''f''(''a'' − ''x'') and ''f''(''x''). It is a special case of a functional equation, and it is very common in the literature to use the term "functional equation" when "reflection formula" is meant. Reflection formulas are useful for numerical computation of special functions. In effect, an approximation that has greater accuracy or only converges on one side of a reflection point (typically in the positive half of the complex plane) can be employed for all arguments. Known formulae The even and odd functions satisfy by definition simple reflection relations around ''a'' = 0. For all even functions, :f(-x) = f(x), and for all odd functions, :f(-x) = -f(x). A famous relationship is Euler's reflection formula :\Gamma(z)\Gamma(1-z) = \frac, \qquad z \not\in \mathbb Z for the gamma function \Gamma(z), due to Leonhard Euler. There is also a refle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Polynomials Of The Second Kind
The Bernoulli polynomials of the second kind , also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating function: : \frac= \sum_^\infty z^n \psi_n(x) ,\qquad , z, -1 and :\gamma=\sum_^\infty\frac\Big\, \quad a>-1 where is Euler's constant. Furthermore, we also have : \Psi(v)= \frac\left\,\qquad \Re(v)>-a, where is the gamma function. The Hurwitz and Riemann zeta functions may be expanded into these polynomials as follows : \zeta(s,v)= \frac + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom (k+v)^ and : \zeta(s)= \frac + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom (k+1)^ and also : \zeta(s) =1 + \frac + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom (k+2)^ The Bernoulli polynomials of the second kind are also involved in the following relationship : \big(v+a-\tfrac\big)\zeta(s,v) = -\frac + \zeta(s-1,v) + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom (k+v)^ between the zeta functions, as w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
