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P ADIC-GAMMA-FUNCTION

  • P-adic gamma function
  • In mathematics, the p-adic gamma function Γp is a function of a p-adic variable analogous to the gamma function. It was first explicitly defined by Morita

    P-adic gamma function

    P-adic_gamma_function

  • Wilson's theorem
  • Theorem on prime numbers

    one to define the p-adic gamma function. Gauss proved that ∏ k = 1 gcd ( k , m ) = 1 m − 1 k   ≡ { − 1 ( mod m ) if  m = 4 , p α , 2 p α 1 ( mod m ) otherwise

    Wilson's theorem

    Wilson's_theorem

  • Gamma function
  • Extension of the factorial function

    approximation Multiple gamma function Multivariate gamma function p-adic gamma function Pochhammer k-symbol Polygamma function q-gamma function Ramanujan's master

    Gamma function

    Gamma function

    Gamma_function

  • Factorial
  • Product of numbers from 1 to n

    that is close to their values to be zero everywhere. Instead, the p-adic gamma function provides a continuous interpolation of a modified form of the factorial

    Factorial

    Factorial

  • Gamma (disambiguation)
  • Topics referred to by the same term

    probability distribution function Gamma function (Γ), a mathematical function P-adic gamma function (Γ), a mathematical function Feferman–Schütte ordinal Γ0 Typing

    Gamma (disambiguation)

    Gamma_(disambiguation)

  • Valuation (algebra)
  • Function in algebra

    {\displaystyle \Gamma =\mathbb {Z} ,} but stronger in general. An elementary example is the p-adic valuation νp associated to a prime integer p, on the rational

    Valuation (algebra)

    Valuation_(algebra)

  • Gross–Koblitz formula
  • Expresses a Gauss sum using a product of values of the p-adic gamma function

    product of values of the p-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Davenport

    Gross–Koblitz formula

    Gross–Koblitz_formula

  • Hasse–Davenport relation
  • Two identities for Gauss sums

    {k-1}{2}}\;k^{1/2-kz}\;\Gamma (kz).\,\!} In fact the Hasse–Davenport product relation follows from the analogous multiplication formula for p-adic gamma functions together

    Hasse–Davenport relation

    Hasse–Davenport_relation

  • Theta function
  • Special functions of several complex variables

    define the Theta functions over other fields where the exponential function might not be everywhere defined, such as fields of p-adic numbers. The Jacobi

    Theta function

    Theta function

    Theta_function

  • Bernard Dwork
  • American mathematician

    an American mathematician, known for his application of p-adic analysis to local zeta functions, and in particular for a proof of the first part of the

    Bernard Dwork

    Bernard_Dwork

  • Multiplication theorem
  • Identity obeyed by many special functions related to the gamma function

    finite form typically occurs only for the gamma and related functions, for which the identity follows from a p-adic relation over a finite field. For example

    Multiplication theorem

    Multiplication_theorem

  • Iwasawa theory
  • Study of objects of arithmetic interest over infinite towers of number fields

    isomorphic to the additive group of p-adic integers for some prime p. (These were called Γ {\displaystyle \Gamma } -extensions in early papers.) Every

    Iwasawa theory

    Iwasawa_theory

  • L-function
  • Meromorphic function on the complex plane

    generalisation of that phenomenon. In that case results have been obtained for p-adic L-functions, which describe certain Galois modules. The statistics of the zero

    L-function

    L-function

    L-function

  • Chowla–Selberg formula
  • Evaluates a certain product of values of the Gamma function at rational values

    there is an analog of the Chowla–Selberg formula for p-adic numbers, involving a p-adic gamma function, called the Gross–Koblitz formula. The Chowla–Selberg

    Chowla–Selberg formula

    Chowla–Selberg_formula

  • Harish-Chandra's c-function
  • Function named after Harish Chandra

    similar c-function for p-adic Lie groups. Macdonald (1968, 1971) and Langlands (1971) found an analogous product formula for the c-function of a p-adic Lie

    Harish-Chandra's c-function

    Harish-Chandra's_c-function

  • Long line (topology)
  • Topological space in mathematics

    \delta <\gamma } is a countable union of p-adic balls, so can be embedded in X γ , {\displaystyle X_{\gamma },} as X γ {\displaystyle X_{\gamma }} with

    Long line (topology)

    Long_line_(topology)

  • Selberg zeta function
  • -\mathbb {N} } . The Ihara zeta function is considered a p-adic (and a graph-theoretic) analogue of the Selberg zeta function. For the case where the surface

    Selberg zeta function

    Selberg_zeta_function

  • Pi
  • Number, approximately 3.14

    \Gamma (n)=(n-1)!} . When the gamma function is evaluated at half-integers, the result contains π. For example, Γ ( 1 2 ) = π {\displaystyle \Gamma {\bigl

    Pi

    Pi

  • Christopher Deninger
  • German mathematician (born 1958)

    on l-adic cohomology of some variety X / Q, while the Euler factor for the infinite place are, according to Serre, products of Gamma functions depending

    Christopher Deninger

    Christopher Deninger

    Christopher_Deninger

  • P-adically closed field
  • Type of field in mathematics

    {\displaystyle \gamma (z)} always has non-negative valuation. The Kochen operator can be thought of as a p-adic (or v-adic) analogue of the square function in the

    P-adically closed field

    P-adically_closed_field

  • Riemann hypothesis
  • Conjecture on zeros of the zeta function

    a p-adic L-function with the eigenvalues of an operator, so can be thought of as an analogue of the Hilbert–Pólya conjecture for p-adic L-functions. Several

    Riemann hypothesis

    Riemann hypothesis

    Riemann_hypothesis

  • Étale cohomology
  • Sheaf cohomology on the étale site

    1960 using p-adic methods), and the remaining conjecture, the analogue of the Riemann hypothesis was proved by Pierre Deligne (1974) using ℓ-adic cohomology

    Étale cohomology

    Étale_cohomology

  • Ramanujan tau function
  • Function studied by Ramanujan

    multiplicative function) τ ( p r + 1 ) = τ ( p ) τ ( p r ) − p 11 τ ( p r − 1 ) {\displaystyle \tau (p^{r+1})=\tau (p)\tau (p^{r})-p^{11}\tau (p^{r-1})} for p {\displaystyle

    Ramanujan tau function

    Ramanujan tau function

    Ramanujan_tau_function

  • Lindemann–Weierstrass theorem
  • Theorem in transcendental number theory

    \mathbb {Q} } , such that | αi |p < 1/p for all i; then the p-adic exponentials expp(α1), . . . , expp(αn) are p-adic numbers that are algebraically independent

    Lindemann–Weierstrass theorem

    Lindemann–Weierstrass theorem

    Lindemann–Weierstrass_theorem

  • Analytic function
  • Type of function in mathematics

    p {\displaystyle \mathbb {Q} _{p}} ∑ n = 0 ∞ a n x n {\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}} converges to an analytic function on the p-adic integers

    Analytic function

    Analytic function

    Analytic_function

  • Harmonic number
  • Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n

    {Ein} (z)=\mathrm {E} _{1}(z)+\gamma +\ln z=\Gamma (0,z)+\gamma +\ln z} where Γ(0, z) is the incomplete gamma function. The harmonic numbers have several

    Harmonic number

    Harmonic number

    Harmonic_number

  • Logarithm
  • Mathematical function, inverse of an exponential function

    (multi-valued) inverse function of the matrix exponential. Another example is the p-adic logarithm, the inverse function of the p-adic exponential. Both are

    Logarithm

    Logarithm

    Logarithm

  • Artin L-function
  • Type of Dirichlet series associated to number field extensions

    product are tied to p-adic places. Using modern notation from the theory of automorphic forms, let's denote Archimedean gamma factors: Γ R ( s ) = π

    Artin L-function

    Artin_L-function

  • Arithmetic function
  • Function whose domain is the positive integers

    exponent. Define the p-adic valuation νp(n) to be the exponent of the highest power of the prime p that divides n. That is, if p is one of the pi then

    Arithmetic function

    Arithmetic_function

  • Harmonic series (mathematics)
  • Divergent sum of positive unit fractions

    \psi (x)={\frac {d}{dx}}\ln {\big (}\Gamma (x){\big )}={\frac {\Gamma '(x)}{\Gamma (x)}}.} Just as the gamma function provides a continuous interpolation

    Harmonic series (mathematics)

    Harmonic_series_(mathematics)

  • Modular forms modulo p
  • Mathematical concept

    theory of complex modular forms and the p-adic theory of modular forms. Modular forms are analytic functions, so they admit a Fourier series. As modular

    Modular forms modulo p

    Modular_forms_modulo_p

  • Faltings' theorem
  • Curves of genus > 1 over the rationals have only finitely many rational points

    Vojta's proof. Brian Lawrence and Akshay Venkatesh gave a proof based on p-adic Hodge theory, borrowing also some of the easier ingredients of Faltings'

    Faltings' theorem

    Faltings' theorem

    Faltings'_theorem

  • Pierre Colmez
  • French mathematician (born 1962)

    He works on special values of L-functions and p {\displaystyle p} -adic representations of p {\displaystyle p} -adic groups at the meeting point of Fontaine's

    Pierre Colmez

    Pierre Colmez

    Pierre_Colmez

  • Ramanujan–Petersson conjecture
  • Unsolved problem in mathematics

    {\frac {\Gamma (s)L(s,\tau )}{(2\pi )^{s}}}={\frac {\Gamma (12-s)L(12-s,\tau )}{(2\pi )^{12-s}}}.} From the mulitplicative property of the tau function, the

    Ramanujan–Petersson conjecture

    Ramanujan–Petersson_conjecture

  • Anatoly Karatsuba
  • Russian mathematician (1937–2008)

    a Russian mathematician working in the field of analytic number theory, p-adic numbers and Dirichlet series. For most of his student and professional life

    Anatoly Karatsuba

    Anatoly Karatsuba

    Anatoly_Karatsuba

  • Greek letters used in mathematics, science, and engineering
  • Symbols for constants, special functions

    matching number of a graph the p-adic valuation of a number Ξ {\displaystyle \Xi } represents: the original Riemann Xi function, i.e. Riemann's lower case

    Greek letters used in mathematics, science, and engineering

    Greek_letters_used_in_mathematics,_science,_and_engineering

  • Integral
  • Operation in mathematical calculus

    antiderivatives, the special functions (like the Legendre functions, the hypergeometric function, the gamma function, the incomplete gamma function and so on). Extending

    Integral

    Integral

    Integral

  • Gelfond–Schneider theorem
  • On the transcendence of a large class of numbers

    either rational or transcendental, where log p {\displaystyle \log _{p}} is the p-adic logarithm function. The transcendence of the following numbers follows

    Gelfond–Schneider theorem

    Gelfond–Schneider_theorem

  • Gauss sum
  • Sum in algebraic number theory

    Stickelberger's theorem B. H. Gross and N. Koblitz. Gauss sums and the p-adic Γ-function. Ann. of Math. (2), 109(3):569–581, 1979. Theorem 9.10 in H. L. Montgomery

    Gauss sum

    Gauss_sum

  • 1 + 1 + 1 + 1 + ⋯
  • Divergent series

    rational ratio that diverges both for the real numbers and for all systems of p-adic numbers. In the context of the extended real number line ∑ n = 1 ∞ 1 = +

    1 + 1 + 1 + 1 + ⋯

    1 + 1 + 1 + 1 + ⋯

    1_+_1_+_1_+_1_+_⋯

  • Maass wave form
  • Complex-valued smooth functions of the upper half plane (harmonic analysis topic)

    smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup Γ {\displaystyle \Gamma } of S L

    Maass wave form

    Maass_wave_form

  • Congruence subgroup
  • Matrix group

    product of all non-archimedean completions (all p-adic fields). If Γ ⊂ G ( Q ) {\displaystyle \Gamma \subset \mathbf {G} (\mathbb {Q} )} is an arithmetic

    Congruence subgroup

    Congruence_subgroup

  • Lattice (group)
  • Periodic set of points

    cases of such lattices occur in number theory with K a p-adic field and T {\displaystyle T} the p-adic integers. For a vector space which is also an inner

    Lattice (group)

    Lattice (group)

    Lattice_(group)

  • Motivic L-function
  • invariants of the v-adic realization of the motive. For infinite places, Jean-Pierre Serre gave a recipe in (Serre 1970) for the so-called Gamma factors in terms

    Motivic L-function

    Motivic_L-function

  • Valuation ring
  • Concept in algebra

    fractions are the functions meromorphic on the whole plane. If f does not have a Maclaurin series then 1/f does. Any ring of p-adic integers Z p {\displaystyle

    Valuation ring

    Valuation_ring

  • Perrin number
  • Number sequence 3,0,2,3,2,5,5,7,10,...

    {\begin{array}{c|c|c}n&P(n)&P(-n)\\\hline 0&P(0)&...\\1&P(1)&P(2)-P(0)\\2&P(2)&-P(2)+P(1)+P(0)\\3&P(1)+P(0)&P(2)-P(1)\\4&P(2)+P(1)&P(1)-P(0)\\5&P(2)+P(1)+P(0)&-P

    Perrin number

    Perrin number

    Perrin_number

  • Nu (Greek)
  • Thirteenth letter in the Greek alphabet

    "vega". The reciprocal of 1 plus the interest rate in finance. The p-adic valuation or p-adic order of a number. Physics: Kinematic viscosity in fluid mechanics

    Nu (Greek)

    Nu_(Greek)

  • Stirling numbers of the first kind
  • Count of permutations by cycles

    Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. There also exist more complicated expressions for the zeta-functions involving the Stirling numbers

    Stirling numbers of the first kind

    Stirling_numbers_of_the_first_kind

  • Motive (algebraic geometry)
  • Structure in algebraic geometry

    Betti cohomology, l-adic cohomology, the number of points over any finite field, and in multiplicative notation for local zeta-functions. The general idea

    Motive (algebraic geometry)

    Motive_(algebraic_geometry)

  • Cantor space
  • Topological space

    f\in L^{1}(\Delta ,\mu )} the function f ^ ( γ ) {\displaystyle {\hat {f}}(\gamma )} of a character γ {\displaystyle \gamma } given by f ^ ( γ ) = ∫ Δ f

    Cantor space

    Cantor_space

  • Gregory coefficients
  • 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

    Gregory_coefficients

  • Automorphic form
  • Type of generalization of periodic functions in Euclidean space

    < G {\displaystyle \Gamma <G} of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to

    Automorphic form

    Automorphic_form

  • Smith–Minkowski–Siegel mass formula
  • reduced mod p r. Some authors state the mass formula in terms of the p-adic density α p ( f ) = N ( p r ) p r n ( n − 1 ) / 2 = p s ( n + 1 ) / 2 m p ( f )

    Smith–Minkowski–Siegel mass formula

    Smith–Minkowski–Siegel_mass_formula

  • Lattice (discrete subgroup)
  • Discrete subgroup in a locally compact topological group

    local fields of characteristic 0, for example the p-adic fields Q p {\displaystyle \mathbb {Q} _{p}} . There is an arithmetic construction similar to

    Lattice (discrete subgroup)

    Lattice (discrete subgroup)

    Lattice_(discrete_subgroup)

  • Selberg trace formula
  • Mathematical theorem

    compact topological field with ultrametric norm, so a finite extension of the p-adic numbers Qp or of the formal Laurent series Fq((T)); they also handle the

    Selberg trace formula

    Selberg_trace_formula

  • Potts model
  • Model in statistical mechanics generalizing the Ising model

    noting that the string of values corresponds to a q-adic number, however the natural topology of the q-adic numbers is finer than the above product topology

    Potts model

    Potts_model

  • Langlands–Shahidi method
  • Representations of p-adic groups: Applications involving Harish-Chandra μ functions (from the Plancherel formula) and to complementary series of p-adic reductive

    Langlands–Shahidi method

    Langlands–Shahidi_method

  • Algebraic independence
  • Set without nontrivial polynomial equalities

    e^{\pi }} , and Γ ( 1 / 4 ) {\displaystyle \Gamma (1/4)} , where Γ {\displaystyle \Gamma } is the gamma function, are algebraically independent over Q {\displaystyle

    Algebraic independence

    Algebraic_independence

  • Eisenstein–Kronecker number
  • Special numbers in mathematics

    congruences that can be used in the construction of two-variable p-adic L-functions. They are related to critical L-values of Hecke characters. When A

    Eisenstein–Kronecker number

    Eisenstein–Kronecker_number

  • Cyclotomic polynomial
  • Irreducible polynomial whose roots are nth roots of unity

    over the p-adic integers, since Hensel's lemma allows lifting a factorization over the field with p elements to a factorization over the p-adic integers

    Cyclotomic polynomial

    Cyclotomic_polynomial

  • Metric space
  • Mathematical space with a notion of distance

    graphs may be viewed as metric spaces. In abstract algebra, the field of p-adic numbers is the completion of the field of rational numbers with respect

    Metric space

    Metric space

    Metric_space

  • List of unsolved problems in mathematics
  • Gan–Gross–Prasad conjecture: a restriction problem in representation theory of real or p-adic Lie groups. Greenberg's conjectures Hermite's problem: is it possible, for

    List of unsolved problems in mathematics

    List_of_unsolved_problems_in_mathematics

  • Baum–Connes conjecture
  • Conjecture linking two mathematical areas

    connected Lie groups or virtually connected Lie groups. Discrete subgroups of p-adic groups. Bolic groups (a certain generalization of hyperbolic groups). Groups

    Baum–Connes conjecture

    Baum–Connes conjecture

    Baum–Connes_conjecture

  • *-autonomous category
  • Symmetric monoidal closed category equipped with a dualizing object

    Drinfeld (2013) mention that the bounded derived category of constructible l-adic sheaves on an algebraic variety has this property. Further examples include

    *-autonomous category

    *-autonomous_category

  • Differentiable manifold
  • Manifold upon which it is possible to perform calculus

    be approached in a coordinate-independent manner using the Ip-adic filtration on OM,p. The tangent bundle (or more precisely its sheaf of sections) can

    Differentiable manifold

    Differentiable manifold

    Differentiable_manifold

  • Mersenne prime
  • Prime number of the form 2^n – 1

    about e γ ⋅ log 2 ⁡ ( 10 ) ≈ 5.92 {\displaystyle e^{\gamma }\cdot \log _{2}(10)\approx 5.92} primes p with n decimal digits for which Mp is prime. Here,

    Mersenne prime

    Mersenne_prime

  • Pohlig–Hellman algorithm
  • Algorithm for computing logarithms

    The basic idea of this algorithm is to iteratively compute the p {\displaystyle p} -adic digits of the logarithm by repeatedly "shifting out" all but one

    Pohlig–Hellman algorithm

    Pohlig–Hellman algorithm

    Pohlig–Hellman_algorithm

  • Harmonic map
  • Concept in mathematics

    f^{\gamma }}{\partial x^{j}}}\Gamma (h)_{\beta \gamma }^{\alpha }\circ f.} Its laplacian defines for each α between 1 and n the real-valued function (∆f)α

    Harmonic map

    Harmonic_map

  • Puiseux series
  • Power series with rational exponents

    There is also an analogous result for p-adic closure: if K {\displaystyle K} is a p {\displaystyle p} -adically closed field with respect to a valuation

    Puiseux series

    Puiseux series

    Puiseux_series

  • Hodge theory
  • Mathematical manifold theory

    applied to questions in number theory. In arithmetic situations, the tools of p-adic Hodge theory have given alternative proofs of, or analogous results to,

    Hodge theory

    Hodge_theory

  • Barrett reduction
  • Algorithm in modular arithmetic

    algorithm for polynomial division, by reversing polynomials and using X-adic arithmetic. Montgomery reduction is another similar algorithm. The remainder

    Barrett reduction

    Barrett_reduction

  • Jet (mathematics)
  • Operation in differential geometry

    can be generalized to smooth functions between Banach spaces, analytic functions between real or complex domains, to p-adic analysis, and to other areas

    Jet (mathematics)

    Jet_(mathematics)

  • Subshift of finite type
  • Type of shift space studied in ergodic theory

    (z)=\prod _{\gamma }\left(1-z^{|\gamma |}\right)^{-1}\ } where γ runs over the closed orbits. For subshifts of finite type, the zeta function is a rational

    Subshift of finite type

    Subshift_of_finite_type

  • Quaternion
  • Four-dimensional number system

    efficiently represent Lorentz boosts, and to interpret formulas involving the gamma matrices.[citation needed] For further detail about the geometrical uses

    Quaternion

    Quaternion

    Quaternion

  • Bell number
  • Count of the possible partitions of a set

    function yields the complex integral representation B n = n ! 2 π i e ∫ γ e e z z n + 1 d z . {\displaystyle B_{n}={\frac {n!}{2\pi ie}}\int _{\gamma

    Bell number

    Bell number

    Bell_number

  • Powerful number
  • Numbers whose prime factors all divide the number more than once

    p i β i ) ( ∏ p i γ i ) = ( ∏ p i β i / 2 ) 2 ( ∏ p i γ i / 3 ) 3 {\displaystyle m=\left(\prod p_{i}^{\beta _{i}}\right)\left(\prod p_{i}^{\gamma

    Powerful number

    Powerful number

    Powerful_number

  • Bernoulli number
  • Rational number sequence

    congruent modulo p − 1 to a particular a ≢ 1 mod (p − 1), and so can be extended to a continuous function ζp(s) for all p-adic integers Z p , {\displaystyle

    Bernoulli number

    Bernoulli_number

  • Polynomial ring
  • Algebraic structure

    p q = ∑ γ ∈ I + J ( ∑ α , β ∣ α + β = γ p α q β ) X γ , {\displaystyle pq=\sum _{\gamma \in I+J}\left(\sum _{\alpha ,\beta \mid \alpha +\beta =\gamma

    Polynomial ring

    Polynomial_ring

  • Hahn series
  • Mathematical formal infinite series

    finite field with p elements, this construction gives a (ultra)metrically complete algebraically closed field containing the p-adics, hence a more or less

    Hahn series

    Hahn_series

  • Basel problem
  • Sum of inverse squares of natural numbers

    of S L 2 ( Z p ) {\displaystyle SL_{2}(\mathbb {Z} _{p})} in each finite place, where Z p {\displaystyle \mathbb {Z} _{p}} is the p-adic integers. For

    Basel problem

    Basel problem

    Basel_problem

  • Tensor product of fields
  • Ring produced from two fields

    also to K ⊗ Q Q p , {\displaystyle K\otimes _{\mathbb {Q} }\mathbb {Q} _{p},} where Q {\displaystyle \mathbb {Q} } p is the field of p-adic numbers. This

    Tensor product of fields

    Tensor_product_of_fields

  • Teichmüller space
  • Parametrizes complex structures on a surface

    p-adic Teichmüller theory Inter-universal Teichmüller theory Teichmüller modular form Imayoshi & Taniguchi 1992, p. 14. Imayoshi & Taniguchi 1992, p. 13

    Teichmüller space

    Teichmüller_space

  • Slash (punctuation)
  • Slanting line punctuation mark (/)

    {\displaystyle \mathbb {Z} _{n}} is also notation for the very different ring of n-adic integers). Z / n {\displaystyle \mathbb {Z} /n} is an abbreviation of Z /

    Slash (punctuation)

    Slash_(punctuation)

  • Sheaf (mathematics)
  • Tool to track locally defined data attached to the open sets of a topological space

    of covering. This allowed Grothendieck to define étale cohomology and ℓ-adic cohomology, which eventually were used to prove the Weil conjectures. A category

    Sheaf (mathematics)

    Sheaf_(mathematics)

  • Colossally abundant number
  • Type of natural number

    s(\varepsilon )=\prod _{p\in \mathbb {P} }p^{e_{p}(\varepsilon )}\ } is a colossally abundant number. For every ε the above function has a maximum, but it

    Colossally abundant number

    Colossally abundant number

    Colossally_abundant_number

  • Clifford algebra
  • Algebra based on a vector space with a quadratic form

    property that γ i γ j + γ j γ i = 2 η i j , {\displaystyle \gamma _{i}\gamma _{j}+\gamma _{j}\gamma _{i}=2\eta _{ij},} where η is the matrix of a quadratic

    Clifford algebra

    Clifford_algebra

  • Fuss–Catalan number
  • Type of number in combinatorial mathematics and statistics

    A_{m}(p,r)\equiv {\frac {r}{mp+r}}{\binom {mp+r}{m}}={\frac {r}{m!}}\prod _{i=1}^{m-1}(mp+r-i)=r{\frac {\Gamma (mp+r)}{\Gamma (1+m)\Gamma (m(p-1)+r+1)}}

    Fuss–Catalan number

    Fuss–Catalan_number

  • Index of physics articles (P)
  • O P Q R S T U V W X Y Z P-adic quantum mechanicsfpa P-factor P-form electrodynamics P-nuclei P-process P-type semiconductor P-wave P-wave modulus P. Buford

    Index of physics articles (P)

    Index_of_physics_articles_(P)

  • Descendant tree (group theory)
  • ) ≃ P {\displaystyle (3)\qquad R\simeq R/\gamma _{c+1}(R)\to R/\gamma _{c}(R)\to \cdots \to R/\gamma _{c+2-m}(R)\to R/\gamma _{c+1-m}(R)\simeq P} , with

    Descendant tree (group theory)

    Descendant_tree_(group_theory)

  • Associative algebra
  • Ring that is also a vector space or a module

    Let Γ = Gal ⁡ ( k s / k ) = lim ← ⁡ Gal ⁡ ( k ′ / k ) {\displaystyle \Gamma =\operatorname {Gal} (k_{s}/k)=\varprojlim \operatorname {Gal} (k'/k)}

    Associative algebra

    Associative_algebra

  • Duality (mathematics)
  • General concept and operation in mathematics

    holds for a smooth projective variety over a separably closed field, using l-adic cohomology with Qℓ-coefficients instead. This is further generalized to possibly

    Duality (mathematics)

    Duality_(mathematics)

  • Superabundant number
  • Class of natural numbers

    (m)}{m}}<{\frac {\sigma (n)}{n}}} where σ denotes the sum-of-divisors function (i.e., the sum of all positive divisors of n, including n itself). The

    Superabundant number

    Superabundant_number

  • Practical number
  • Number whose sums of distinct divisors represent all smaller numbers

    practical 1 n ( ∑ p ≤ σ ( n ) + 1 log ⁡ p p − 1 − log ⁡ n ) ∏ p ≤ σ ( n ) + 1 ( 1 − 1 p ) , {\displaystyle c={\frac {1}{1-e^{-\gamma }}}\sum _{n\

    Practical number

    Practical number

    Practical_number

  • History of mathematics
  • the gamma function, arXiv:math/0505125, Bibcode:2005math......5125B Askey, Richard (1980). "Ramanujan's Extensions of the Gamma and Beta Functions". The

    History of mathematics

    History of mathematics

    History_of_mathematics

  • Repunit
  • Numbers that contain only the digit 1

    {b^{n}-1}{b-1}}} is prime (for prime p {\displaystyle p} ) is about e γ p ⋅ log e ⁡ ( | b | ) {\displaystyle {\frac {e^{\gamma }}{p\cdot \log _{e}(|b|)}}} . Although

    Repunit

    Repunit

  • Multiply perfect number
  • Number whose divisors add to a multiple of that number

    e − γ {\displaystyle \log \log n>k\cdot e^{-\gamma }} where γ {\displaystyle \gamma } is Euler's gamma constant. This can be proven using Robin's theorem

    Multiply perfect number

    Multiply perfect number

    Multiply_perfect_number

  • Projective variety
  • Algebraic variety in a projective space

    (0:0:1)\\z\mapsto (1:\wp (z):\wp '(z))\end{cases}}} There is a p-adic analog, the p-adic uniformization theorem. For higher dimensions, the notions of

    Projective variety

    Projective variety

    Projective_variety

  • Complex torus
  • Kind of complex manifold

    Complex Lie group Automorphic function Intermediate Jacobian Elliptic gamma function Mumford, David (2008). Abelian varieties. C. P. Ramanujam, I︠U︡. I. Manin

    Complex torus

    Complex torus

    Complex_torus

  • Glossary of algebraic geometry
  • algebraically closed field k; that is, | D | = P ( Γ ( X , O X ( D ) ) ) {\displaystyle |D|=\mathbf {P} (\Gamma (X,{\mathcal {O}}_{X}(D)))} . There is a bijection

    Glossary of algebraic geometry

    Glossary_of_algebraic_geometry

  • Compact group
  • Topological group with compact topology

    carry the structure of a manifold, examples are the additive group Zp of p-adic integers, and constructions from it. In fact any profinite group is a compact

    Compact group

    Compact group

    Compact_group

AI & ChatGPT searchs for online references containing P ADIC-GAMMA-FUNCTION

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P ADIC-GAMMA-FUNCTION

  • Adin
  • Boy/Male

    Hebrew

    Adin

    Attractive; handsome; pleasure given. Adin was a biblical exile who returned to Israel from Babylon.

    Adin

  • Tamma
  • Girl/Female

    Hebrew

    Tamma

    Without flaw.

    Tamma

  • ARIC
  • Male

    English

    ARIC

    Variant spelling of English Eric, ARIC means "ever-ruler."

    ARIC

  • FÜLÖP
  • Male

    Hungarian

    FÜLÖP

    Hungarian form of English Philip, FÜLÖP means "lover of horses."

    FÜLÖP

  • Gemma
  • Girl/Female

    African, American, Australian, British, Chinese, Christian, Danish, Dutch, English, French, German, Irish, Italian, Jamaican, Latin

    Gemma

    Jewel; Precious Stone; Gem

    Gemma

  • Gamya
  • Girl/Female

    Hindu, Indian, Kannada, Telugu

    Gamya

    Beautiful; A Destiny

    Gamya

  • Tamma
  • Girl/Female

    Australian, French, Hebrew

    Tamma

    Without Flaw; Palm Tree; Perfect

    Tamma

  • GEMMA
  • Female

    English

    GEMMA

    Italian name GEMMA means "precious stone."

    GEMMA

  • Damma
  • Girl/Female

    Gujarati, Hindu, Indian

    Damma

    The Soothing Voice

    Damma

  • Adiy
  • Boy/Male

    Indian

    Adiy

    A companion of the prophet, Also the name of the son of Hatim tiay known for his generosity, Also the son of Thabit had this name

    Adiy

  • ADIN
  • Male

    English

    ADIN

    Anglicized form of Hebrew Adiyn, ADIN means "dainty, delicate." In the bible, this is the name of an ancestor of a family of exiles who returned with Zerubbabel.

    ADIN

  • Samma
  • Girl/Female

    Arabic, Indian, Kashmiri

    Samma

    Beautiful Sky

    Samma

  • Kamma
  • Girl/Female

    Danish, Indian, Latin, Sanskrit, Swedish

    Kamma

    Loveable; Desire

    Kamma

  • Adiv
  • Boy/Male

    Indian

    Adiv

    Pleasant

    Adiv

  • Amma
  • Girl/Female

    Norse

    Amma

    Grandmother.

    Amma

  • Amma
  • Boy/Male

    African, British, English, Indian

    Amma

    Mother; God-like

    Amma

  • Gemma
  • Girl/Female

    French Latin Italian

    Gemma

    Jewel.

    Gemma

  • Amma
  • Boy/Male

    Indian

    Amma

    Supreme god.

    Amma

  • ADI
  • Female

    English

    ADI

    (עֲדִי) Hebrew unisex name ADI means "my ornament" or "my witness."

    ADI

  • ALIC
  • Male

    English

    ALIC

    Short form of English Alexander, ALIC means "defender of mankind."

    ALIC

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Online names & meanings

  • Shivadev | ஷிவதேவ
  • Boy/Male

    Tamil

    Shivadev | ஷிவதேவ

    Lord of prosperity

  • Wilcher
  • Surname or Lastname

    English

    Wilcher

    English : variant of Wiltshire.

  • Fannia
  • Girl/Female

    English

    Fannia

    free;.

  • Brenton
  • Boy/Male

    American, Australian, British, Celtic, Chinese, Christian, English, Jamaican

    Brenton

    Hilltop; Mount; Variant of Brent; Settlement Associated with Bryni; Fire; Flame

  • WEN
  • Male

    Chinese

    WEN

    genial.

  • Diggory
  • Boy/Male

    British, Christian, English, French

    Diggory

    Astray

  • Shvetanshu | ஷ்வேதாந்ஷு
  • Boy/Male

    Tamil

    Shvetanshu | ஷ்வேதாந்ஷு

    The Moon

  • Philip
  • Boy/Male

    Christian & English(British/American/Australian)

    Philip

    Horse Lover

  • Bink
  • Surname or Lastname

    English

    Bink

    English : topographic name for someone living by a bink, a northern dialect term for a flat raised bank of earth or a shelf of flat stone suitable for sitting on. The word is a northern form of modern English bench.Variant of Polish Binek, itself a variant of Bieniek.

  • Saalif
  • Boy/Male

    Arabic, Muslim

    Saalif

    Name of a Sahabi

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  • Gemmae
  • pl.

    of Gemma

  • Gadic
  • a.

    Pertaining to, or derived from, the cod (Gadus); -- applied to an acid obtained from cod-liver oil, viz., gadic acid.

  • Mammae
  • pl.

    of Mamma

  • Mama
  • n.

    See Mamma.

  • Amic
  • a.

    Related to, or derived, ammonia; -- used chiefly as a suffix; as, amic acid; phosphamic acid.

  • Gamba
  • n.

    A viola da gamba.

  • Gummata
  • pl.

    of Gumma

  • Gamma
  • n.

    The third letter (/, / = Eng. G) of the Greek alphabet.

  • Mam
  • n.

    Mamma.