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Meromorphic function
In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers C {\displaystyle \mathbb {C} } defined as the (m +
Polygamma_function
In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor
Balanced_polygamma_function
Extension of the factorial function
gamma function Multivariate gamma function p-adic gamma function Pochhammer k-symbol Polygamma function q-gamma function Ramanujan's master theorem Spouge's
Gamma_function
Identity obeyed by many special functions related to the gamma function
The polygamma identities can be used to obtain a multiplication theorem for harmonic numbers. The Hurwitz zeta function generalizes the polygamma function
Multiplication_theorem
Mathematical function
(z)={\frac {\Gamma '(z)}{\Gamma (z)}}.} It is the first of the polygamma functions. This function is strictly increasing and strictly concave on ( 0 , ∞ ) {\displaystyle
Digamma_function
Transcendental single-variable function
tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function. The Clausen function of order 2 – often
Clausen_function
Mathematical function
In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by ψ 1 ( z ) = d 2 d z 2 ln
Trigamma_function
Analytic function in mathematics
where ψ {\displaystyle \psi } and γ {\displaystyle \gamma } are the polygamma function and Euler's constant, respectively, as well as ∑ n = 1 ∞ ζ ( 2 n )
Riemann_zeta_function
Concept in mathematics
s}}\right|_{s=a},\ \ \zeta (s,q)=\sum _{k=0}^{\infty }(k+q)^{-s}} Definition via polygamma function: K ( z ) = exp [ ψ ( − 2 ) ( z ) + z 2 − z 2 − z 2 ln 2 π ] {\displaystyle
K-function
Special mathematical function
Also the series representation of Dirichlet beta function can be formed in terms of the polygamma function β ( s ) = 1 2 s ∑ n = 0 ∞ ( − 1 ) n ( n + 1 2
Dirichlet_beta_function
Riemann zeta function. Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. ψ n ( z ) {\displaystyle \psi _{n}(z)} is a polygamma function. Li s (
List_of_mathematical_series
analogue. Digamma function, Polygamma function Incomplete beta function Incomplete gamma function K-function Multivariate gamma function: A generalization
List of mathematical functions
List_of_mathematical_functions
Probability distribution
moment-generating function is MX(t)=𝔼[etX]=(1−p+pet)n. Usually the mode of a binomial B(n, p) distribution is equal to ⌊(n+1)p⌋, where ⌊⋅⌋ is the floor function. However
Binomial_distribution
Topics referred to by the same term
Chebyshev function ψ ( x ) {\displaystyle \psi (x)} the polygamma function ψ m ( z ) {\displaystyle \psi ^{m}(z)} or its special cases the digamma function ψ
Psi_function
Number of subsets of a given size
previous generating function after the substitution x → x y {\displaystyle x\to xy} . A symmetric exponential bivariate generating function of the binomial
Binomial_coefficient
Penultimate letter in the Greek alphabet
S2CID 193082268. Weisstein, Eric W. "Polygamma Function". mathworld.wolfram.com. Retrieved 2025-02-08. A special function mostly commonly denoted ψ_n(z), ψ^((n))(z)
Psi_(Greek)
Inverse of the gamma function
{1}{z^{4}}}\right)\,,} where ψ ( n ) ( x ) {\displaystyle \psi ^{(n)}(x)} is the polygamma function. This can be rigorously justified by the Lagrange inversion theorem
Inverse_gamma_function
Numerical computation of special functions
gamma function Γ ( z ) {\textstyle \Gamma (z)} , due to Leonhard Euler. There is also a reflection formula for the general n-th order polygamma function ψ(n)(z)
Reflection_formula
Special function in mathematics
uniform electric field. The Hurwitz zeta function with a positive integer m is related to the polygamma function: ψ ( m ) ( z ) = ( − 1 ) m + 1 m ! ζ (
Hurwitz_zeta_function
Triangular array of the binomial coefficients
convolving the distribution function for a random variable with itself corresponds to calculating the distribution function for a sum of n independent
Pascal's_triangle
{t}{1-t}}\right]} which holds for |t| < 2. Here, ψ is the digamma function and ψ(m) is the polygamma function. Many series involving the binomial coefficient may be
Rational_zeta_series
Mathematical function
the polygamma function of order 2 k {\displaystyle 2k} . The Riemann–Siegel theta function is of interest in studying the Riemann zeta function, since
Riemann–Siegel_theta_function
Graphical aid for deriving some concepts in combinatorics
objects are not distinguished). This is represented by the generating function 1+1x+1x2+1x3+…=1+x+x2+x3+…=11−x. The series is a geometric series, and
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Algebraic expansion of powers of a binomial
{n}{k}}={\frac {n!}{k!\;(n-k)!}},} which is defined in terms of the factorial function n!. Equivalently, this formula can be written ( n k ) = n ( n − 1 ) ⋯ (
Binomial_theorem
Symbols for constants, special functions
reciprocal Fibonacci constant the second Chebyshev function in number theory the polygamma function in mathematics the supergolden ratio In circadian physiology
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Product of consecutive factorial numbers
continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function as s f ( n ) = G ( n + 2 ) {\displaystyle
Superfactorial
Type of permutation of a set of elements
Dn=[n!e]=⌊n!e+12⌋for n≥1, where [x] is the nearest integer function and ⌊x⌋ is the floor function. Other related formulas include Dn=⌊n!+1e⌋for n≥1,Dn=⌊(e+e−1)n
Derangement
Probability distribution
(\alpha )} is strictly concave, by using inequality properties of the polygamma function. Finding the maximum with respect to α by taking the derivative and
Gamma_distribution
Number, approximately 0.916
[citation needed] G appears in values of the second polygamma function, also called the trigamma function, at fractional arguments: ψ 1 ( 1 4 ) = π 2 + 8
Catalan's_constant
Probability distribution
the trigamma function, denoted ψ1(α), is the second of the polygamma functions, and is defined as the derivative of the digamma function: ψ 1 ( α ) =
Beta_distribution
Upper bound on intersecting set families
through any single point. A signed set consists of a set together with sign function that maps each element to { 1 , − 1 } {\displaystyle \{1,-1\}} . Two signed
Erdős–Ko–Rado_theorem
Probability distribution
binomial coefficient that appears in the formula for the probability mass function of the distribution can be written more simply with negative numbers. Imagine
Negative binomial distribution
Negative_binomial_distribution
Mathematical theorem on convolved binomial coefficients
b;c;1)=Γ(c)Γ(c−a−b)Γ(c−a)Γ(c−b) where 2F1 is the hypergeometric function and Γ(n+1)=n! is the gamma function. One regains the Chu–Vandermonde identity by taking a = −n
Vandermonde's_identity
Branch of discrete mathematics
combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic
Combinatorics
Probability distribution
1)\psi ^{0}(k/2))} where ψ 0 ( z ) {\displaystyle \psi ^{0}(z)} is the polygamma function. We find the large n=k+1 approximation of the mean and variance of
Chi_distribution
Topics referred to by the same term
Melchior Islands, Antarctica Chebyshev function Dedekind psi function Digamma function Polygamma functions Stream function, in two-dimensional flows Polar tangential
Psi
Family of probability distributions often used to model tails or extreme values
parameter ξ {\displaystyle \xi } only through the polygamma function of order 1 (also called the trigamma function): Var [ Y ] = { ψ ′ ( 1 ) − ψ ′ ( − 1 / ξ
Generalized Pareto distribution
Generalized_Pareto_distribution
Mathematical operation in calculus
needed] The digamma function, and by extension the polygamma function, is defined in terms of the logarithmic derivative of the gamma function. Generalizations
Logarithmic_derivative
Special mathematical function
{z}{2^{s}}}\Phi (-z^{2},s,{\tfrac {1}{2}})} The polygamma functions for positive integers n: ψ ( n ) ( α ) = ( − 1 ) n + 1 n ! Φ ( 1
Lerch_transcendent
Theorem on prime numbers
practical value. Wilson's theorem allows one to define the p-adic gamma function. Gauss proved that ∏ k = 1 gcd ( k , m ) = 1 m − 1 k ≡ { − 1 ( mod m
Wilson's_theorem
Mathematical series
(\alpha -k+1)}{k!}}.} The binomial series is the MacLaurin series for the function f ( x ) = ( 1 + x ) α {\displaystyle f(x)=(1+x)^{\alpha }} . It converges
Binomial_series
Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
integer, and m > 1 {\displaystyle m>1} integer or not, we have from polygamma functions: H q / p , m = ζ ( m ) − p m ∑ k = 1 ∞ 1 ( q + p k ) m {\displaystyle
Harmonic_number
Arithmetical function
In number theory, the Kempner function S ( n ) {\displaystyle S(n)} is defined for a given positive integer n {\displaystyle n} to be the smallest number
Kempner_function
Generalization of the binomial theorem to other polynomials
Kummer's theorem. By Stirling's approximation, or equivalently the log-gamma function's asymptotic expansion, log ( k n n , n , ⋯ , n ) = k n log ( k ) +
Multinomial_theorem
Type of polynomial sequence
factorials" is defined by (x)n=x(x−1)(x−2)⋅⋯⋅(x−n+1). (In the theory of special functions, this same notation denotes upper factorials, but this present usage is
Binomial_type
Probability distribution on a hyper-sphere of arbitrary dimension
where ψ ′ = ψ ( 1 ) {\displaystyle \psi '=\psi ^{(1)}} is the first polygamma function. The variances decrease, the distributions of all three variables
Von_Mises–Fisher_distribution
Recurrence relations of binomial coefficients in Pascal's triangle
(rising/falling) · Digamma function · Polygamma function · Multivariate gamma function · Hypergeometric series · Hypergeometric function identities Factorials
Hockey-stick_identity
(rising/falling) · Digamma function · Polygamma function · Multivariate gamma function · Hypergeometric series · Hypergeometric function identities Factorials
Alternating_factorial
Sequence of numbers ((2n) choose (n))
}}=e^{2x}I_{0}(2x),} where I0 is a modified Bessel function of the first kind. The generating function of the squares of the central binomial coefficients
Central_binomial_coefficient
Recursive integer sequence
Stirling's approximation for n ! {\displaystyle n!} , or via generating functions. The only odd Catalan numbers C n {\displaystyle C_{n}} are those
Catalan_number
Transformation of a mathematical sequence
binomial transform to the sequence associated with its ordinary generating function. The binomial transform, T, of a sequence, {an}, is the sequence {sn} defined
Binomial_transform
Mathematical constant
ISSN 0962-8444. JSTOR 52768. Adamchik, Victor S. (1998-12-21). "Polygamma functions of negative order". Journal of Computational and Applied Mathematics
Glaisher–Kinkelin_constant
Number theory expression
{n}{p^{i}}}\right\rfloor ,} where ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } is the floor function. While the sum on the right side is an infinite sum, for any particular
Legendre's_formula
Numbers expressible as integrals of algebraic functions
a complex number that can be expressed as an integral of an algebraic function over an algebraic domain. The periods are a class of numbers which includes
Period_(number_theory)
Multivariate generalization of the gamma function
_{p}(a)}{\partial a}}=\sum _{i=1}^{p}\psi (a+(1-i)/2),} and the general polygamma function as ψ p ( n ) ( a ) = ∂ n log Γ p ( a ) ∂ a n = ∑ i = 1 p ψ ( n )
Multivariate_gamma_function
(also falling, lower, rising, upper factorials) Poisson distribution Polygamma function Primorial Proof of Bertrand's postulate Sierpinski triangle Star of
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Operation on formal power series
other series for the zeta-function-related cases of the Legendre chi function, the polygamma function, and the Riemann zeta function include χ 1 ( z ) = ∑
Generating function transformation
Generating_function_transformation
Numbering of combinations of items
(rising/falling) · Digamma function · Polygamma function · Multivariate gamma function · Hypergeometric series · Hypergeometric function identities Factorials
Combinatorial_number_system
Mathematical result on arithmetic properties of binomial coefficients
(rising/falling) · Digamma function · Polygamma function · Multivariate gamma function · Hypergeometric series · Hypergeometric function identities Factorials
Star_of_David_theorem
Summation formula
Here the left-hand side is equal to ψ(1)(z), namely the first-order polygamma function defined by ψ ( 1 ) ( z ) = d 2 d z 2 ln Γ ( z ) ; {\displaystyle
Euler–Maclaurin_formula
that the magnitude square functions of low-pass and high-pass binomial-QMF filters are the unique maximally flat functions in a two-band PR-QMF design
Binomial_QMF
Generalization of the binomial distribution
non-negative integers x1, ..., xk. The probability mass function can be expressed using the gamma function as: f ( x 1 , … , x k ; p 1 , … , p k ) = Γ ( ∑ i
Multinomial_distribution
Family of polynomials
empty products. Although the formula at first appears to be a rational function, it actually is a polynomial, because the division is exact in Z[q]. All
Gaussian_binomial_coefficient
Product of the first "n" prime numbers
# {\displaystyle p_{n}\#} ", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying
Primorial
Inverse of a finite difference
the Gauss Pi function, Π ( x ) = Γ ( x + 1 ) {\displaystyle \Pi (x)=\Gamma (x+1)} . The digamma and more generally polygamma functions are obtained by
Indefinite_sum
Mathematical functions
_{k=0}^{n-1}(x-k).} The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product
Falling_and_rising_factorials
Name for several different families of probability distributions
function, while ψ ′ = ψ ( 1 ) {\displaystyle \psi '=\psi ^{(1)}} is its first derivative, also known as the trigamma function, or the first polygamma
Generalized logistic distribution
Generalized_logistic_distribution
Number computed as a product of powers
continuously interpolated by the gamma function, the hyperfactorials can be continuously interpolated by the K-function as K ( n + 1 ) = H ( n ) {\displaystyle
Hyperfactorial
Inverse of a finite difference/the antidifference for various functions
negative integer exponents, the indefinite sum is closely related to the polygamma function: ∑ x 1 x a = ( − 1 ) a − 1 ψ ( a − 1 ) ( x ) ( a − 1 ) ! + C , a ∈
List_of_indefinite_sums
Generalized chain rule in calculus
polynomial. In case g(x) is a cumulant-generating function, then f(g(x)) is a moment-generating function, and the polynomial in various derivatives of g
Faà_di_Bruno's_formula
Generalization of the mathematical factorial
mathematics, Bhargava's factorial function, or simply the Bhargava factorial, is a generalization of the factorial function developed by Manjul Bhargava as
Bhargava_factorial
Conjecture in combinatorial number theory
(rising/falling) · Digamma function · Polygamma function · Multivariate gamma function · Hypergeometric series · Hypergeometric function identities Factorials
Singmaster's_conjecture
Type of prime number
{\displaystyle (p-1)!+1} , where " ! {\displaystyle !} " denotes the factorial function; compare this with Wilson's theorem, which states that every prime p {\displaystyle
Wilson_prime
Numeral system in combinatorics
permutations. It is also called factorial base, although factorials do not function as base, but as place value of digits. By converting a number less than
Factorial_number_system
Numerical method for calculating the gamma function
mathematics, the Lanczos approximation is a method for computing the gamma function numerically, published by Cornelius Lanczos in 1964. It is a practical
Lanczos_approximation
Sum of the inverses of the positive cubes
representations via the known integral formulas for the gamma and polygamma functions. Apéry's constant is related to the following continued fraction:
Apéry's_constant
Rational numbers in a reciprocal logarithm
known that the zeta function, the gamma function, the polygamma functions, the Stieltjes constants and many other special functions and constants may be
Gregory_coefficients
Theorem in p-adic analysis
theorem, introduced by Kurt Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special polynomials. This series is referred
Mahler's_theorem
Solved prime-number problem
factor of p to the numerator.) An upper bound is supplied for the primorial function, n#=∏p≤np, where the product is taken over all prime numbers p less than
Proof_of_Bertrand's_postulate
Formula in mathematics
(rising/falling) · Digamma function · Polygamma function · Multivariate gamma function · Hypergeometric series · Hypergeometric function identities Factorials
Trinomial_expansion
Uniqueness theorem in complex analysis
Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers
Carlson's_theorem
Expectation or average of the falling factorial of a random variable
integer-valued random variables, and arise in the use of probability-generating functions to derive the moments of discrete random variables. Factorial moments
Factorial_moment
Arrangement of trinomial coefficients
C, respectively. Shorthand is normally used to express combinatorial functions in the following "choose" format (which is read as "4 choose 4, 0, 0"
Pascal's_pyramid
Mathematical sequences in combinatorics
be found on the page for Stirling numbers and exponential generating functions. Another, infrequent notation is s 1 ( n , k ) {\displaystyle s_{1}(n
Stirling_number
Mathematical identity involving sums of binomial coefficients
(rising/falling) · Digamma function · Polygamma function · Multivariate gamma function · Hypergeometric series · Hypergeometric function identities Factorials
Abel's_binomial_theorem
Recursive mathematical formula
factorial of a positive integer n {\displaystyle n} is a quickly growing function of n {\displaystyle n} defined by iterating the process of exponentiation
Exponential_factorial
)^{2}}}=\sum _{k=0}^{n}{n \choose 2k}{2k \choose k}.} Their generating function is 1 + x + 3 x 2 + 7 x 3 + 19 x 4 + … = 1 ( 1 + x ) ( 1 − 3 x ) . {\displaystyle
Trinomial_triangle
Number that is the product of factorials
polynomial of n {\displaystyle n} , but more slowly than any exponential function of n {\displaystyle n} . More precisely, for every ε > 0 {\displaystyle
Jordan–Pólya_number
Integer sequence
proportionality that oscillates between 0.812556... and 1, periodically as a function of log n. The first 2i values in Gould's sequence may be constructed by
Gould's_sequence
(2001), 13-20. A. Laforgia, P. Natalini, Exponential, gamma and polygamma functions: Simple proofs of classical and new inequalities, J. Math. Anal.
Gautschi's_inequality
the generating function identity f ( x ) = g ( log ( 1 + x ) ) {\displaystyle f(x)=g(\log(1+x))} . Binomial transform Generating function transformation
Stirling_transform
Triangular array of natural numbers
exactly k blocks, we use the Narayana number N(n,k). The generating function for the Narayana numbers is ∑n=1∞∑k=1nN(n,k)zntk−1=1−z(t+1)−1−2z(t+1)+z2(t−1)22tz
Narayana_number
=-1/2} generates Stirling's interpolation polynomials. Given an analytic function f ( z ) {\displaystyle f(z)} , define the moving difference of f as L n
Difference_polynomials
Mathematical integral
sometimes called Rice's method, relates the nth forward difference of a function to a contour integral on the complex plane. It commonly appears in the
Nørlund–Rice_integral
generating function perspective. For instance, the integrand is usually a rational function, and the sum of the residues of a rational function is zero,
Egorychev_method
Type of number in combinatorial mathematics and statistics
"Densities of the Raney distributions" paper, let the ordinary generating function with respect to the index m be defined as follows: B p , r ( z ) := ∑ m
Fuss–Catalan_number
Mathematics term
(rising/falling) · Digamma function · Polygamma function · Multivariate gamma function · Hypergeometric series · Hypergeometric function identities Factorials
Lozanić's_triangle
Infinite matrices with Pascal's triangle as elements
matrices G. Helms Gauss-matrix Weisstein, Eric W. Gaussian-function Weisstein, Eric W. Erf-function Weisstein, Eric W. "Hermite Polynomial". Hermite-polynomials
Pascal_matrix
Swedish mathematician and politician (1814–1886)
curious that some of Malmsten's integrals lead to the gamma- and polygamma functions of a complex argument, which are not often encountered in analysis
Carl_Johan_Malmsten
American mathematician
1090/s0002-9947-1924-1501261-5. MR 1501261. Davis, H. T. (1935). "An extension to polygamma functions of a theorem of Gauss". Bull. Amer. Math. Soc. 41 (4): 243–247. doi:10
Harold_Thayer_Davis
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POLYGAMMA FUNCTION
POLYGAMMA FUNCTION
POLYGAMMA FUNCTION
POLYGAMMA FUNCTION
POLYGAMMA FUNCTION
POLYGAMMA FUNCTION
POLYGAMMA FUNCTION
POLYGAMMA FUNCTION
POLYGAMMA FUNCTION
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