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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
Algebraic expansion of powers of a binomial
the formula (nk)=n!k!(n−k)!, which is defined in terms of the factorial function n!. Equivalently, this formula can be written (nk)=n(n−1)⋯(n−k+1)k(k−1
Binomial_theorem
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
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
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
English-born American mathematician
English-born American mathematician, known for the Kempner function and the Kempner series. Aubrey Kempner received in 1911 his PhD with the dissertation
Aubrey_J._Kempner
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
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
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
Product of numbers from 1 to n
is a prime number. For any given integer x {\displaystyle x} , the Kempner function of x {\displaystyle x} is given by the smallest n {\displaystyle n}
Factorial
Recurrence relations of binomial coefficients in Pascal's triangle
Narayana numbers · Lah number · Gould's sequence · Jordan–Pólya number · Kempner function · Gaussian binomial coefficient Multinomial generalizations Multinomial
Hockey-stick_identity
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
Branch of discrete mathematics
combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic
Combinatorics
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
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
Topics referred to by the same term
Kempner number or Kempner numbers may refer to: The values of the Kempner function, the smallest positive integer m such that n divides m!. The Kempner
Kempner number (disambiguation)
Kempner_number_(disambiguation)
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
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)
Narayana numbers · Lah number · Gould's sequence · Jordan–Pólya number · Kempner function · Gaussian binomial coefficient Multinomial generalizations Multinomial
Alternating_factorial
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 identity involving sums of binomial coefficients
Narayana numbers · Lah number · Gould's sequence · Jordan–Pólya number · Kempner function · Gaussian binomial coefficient Multinomial generalizations Multinomial
Abel's_binomial_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
Mathematical result on arithmetic properties of binomial coefficients
Narayana numbers · Lah number · Gould's sequence · Jordan–Pólya number · Kempner function · Gaussian binomial coefficient Multinomial generalizations Multinomial
Star_of_David_theorem
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
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
Formula in mathematics
Narayana numbers · Lah number · Gould's sequence · Jordan–Pólya number · Kempner function · Gaussian binomial coefficient Multinomial generalizations Multinomial
Trinomial_expansion
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
Hypergeometric series Incomplete beta function Incomplete gamma function Jordan–Pólya number Kempner function Lah number Lanczos approximation Lozanić's
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
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
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
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
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
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
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
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
Conjecture in combinatorial number theory
Narayana numbers · Lah number · Gould's sequence · Jordan–Pólya number · Kempner function · Gaussian binomial coefficient Multinomial generalizations Multinomial
Singmaster's_conjecture
Numbering of combinations of items
Narayana numbers · Lah number · Gould's sequence · Jordan–Pólya number · Kempner function · Gaussian binomial coefficient Multinomial generalizations Multinomial
Combinatorial_number_system
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
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
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
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
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
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
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
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
French mathematician (1842–1891)
the umbral calculus. Lucas is credited as the first to publish the Kempner function. Lucas was also interested in recreational mathematics. He found an
Édouard_Lucas
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
)^{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
the generating function identity f ( x ) = g ( log ( 1 + x ) ) {\displaystyle f(x)=g(\log(1+x))} . Binomial transform Generating function transformation
Stirling_transform
American mayor and businessman (1873–1967)
Cincinnati on January 14, 1873, to Harris Kempner and Eliza Seinsheimer Kempner. Although the newly married Kempners were residing in Galveston, Eliza traveled
Isaac_Herbert_Kempner
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
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
=-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
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
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
Mathematics term
Narayana numbers · Lah number · Gould's sequence · Jordan–Pólya number · Kempner function · Gaussian binomial coefficient Multinomial generalizations Multinomial
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
Cosmological formulas for expanding universe
Table/Planck2018 at "Lambda-CDM model#Parameters") Kempner, Joshua (2022). "KEMPNER Cosmology Calculator". Kempner.net. Retrieved 6 August 2022. KEMP Cosmology
Distance_measure
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
Lithuanian-Israeli poet, writer, and partisan leader (1918–1987)
the Soviet army in July 1944, Kovner, along with his lieutenants Vitka Kempner and Rozka Korczak, commanded a partisan group called the Avengers ("Nokmim"
Abba_Kovner
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
Fad diet, radical treatment of malignant hypertension
Walter Kempner [de] (1903-1997), a German physician and refugee from the Nazis, who was at that time associated with Duke University. Kempner had many
Rice_diet
Mathematical constant; sum of 1 / 2^2^n
The Kempner number is the sum of the series κ := 2 − 2 0 + 2 − 2 1 + 2 − 2 2 + ⋯ = ∑ n ≥ 0 2 − 2 n . {\displaystyle \kappa :=2^{-2^{0}}+2^{-2^{1}}+2^{-2^{2}}+\cdots
Kempner_number
In mathematics, a non-algebraic number
1858, pp. 97–98; Bourbaki 1994, p. 74 Erdős & Dudley 1983 Lambert 1768 Kempner 1916 "Weisstein, Eric W. "Liouville's Constant", MathWorld". Liouville
Transcendental_number
Mathematical sequence
{1}{k!}}\left({\frac {x}{1-x}}\right)^{k}} The n-th derivative of the function e 1 x {\displaystyle e^{\frac {1}{x}}} can be expressed with the Lah numbers
Lah_number
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
L-Functions and Arithmetic. Cambridge University Press. p. 333. ISBN 978-0-521-38619-7. Robert Baillie (2013). "Summing The Curious Series of Kempner and
List of mathematical constants
List_of_mathematical_constants
American animated fantasy musical children's television series
chops, in addition to obtaining her super harness. Geo (voiced by Ethan Kempner in the first three seasons and Juan Mirt in the fourth season) is Milli's
Team_Umizoomi
Animated sports fantasy series
Savarog Ruthie Henshall Arina Ii as Pregrina Freaks, Ondine McKraken Luke Kempner Gregg Lowe as Griffin Wells Omar Malik Shai Matheson as Announcer Joseph
Dragon_Striker
Erectile female sexual organ
117–138; Labuski 2015, p. 19 Shrage & Stewart 2015, pp. 225–229; Schwartz & Kempner 2015, p. 24; Wood 2017, pp. 68–69; Blechner 2017 Rodgers 2003, pp. 92–93;
Clitoris
Fixed number that has received a name
Collection Bogomolny, Alexander. "Square root of 2 is irrational". Aubrey J. Kempner (Oct 1916). "On Transcendental Numbers". Transactions of the American Mathematical
Mathematical_constant
1994 French TV series or program
(Flynn) Original voice: Daniel Lafourcade David Gasman (US dub) Teddy Kempner (UK dub) A walking stick born a Yuk/Krud, but adopted by the Joyces/Verigreens
Insektors
term is a and the common ratio is r then the sum is r/ a (r − 1). The Kempner series is the sum of the reciprocals of all positive integers not containing
List_of_sums_of_reciprocals
Trial
for the Prosecution was Telford Taylor; the chief prosecutor was Robert Kempner. The indictment was filed on 15 November 1947; the hearings lasted from
Ministries_Trial
Set of integers whose sum of reciprocals diverges
expansion does not include the digit 7 is small. Such series are called Kempner series. Any set whose upper asymptotic density is nonzero, is large. The
Large_set_(combinatorics)
American businessman and programmer (born 1984)
York, to Jewish parents Dr. Edward Zuckerberg, a dentist, and Karen (née Kempner), a psychiatrist. He and his three sisters (Arielle, Randi, and Donna)
Mark_Zuckerberg
British author and Holocaust denier (born 1938)
visit to Germany, Irving met Robert Kempner, one of the American prosecutors at the Nuremberg trials. Irving asked Kempner whether the "official record of
David_Irving
Indian-American computational neuroscientist
of Neurobiology at Harvard Medical School and founding faculty in the Kempner Institute for the Study of Natural and Artificial Intelligence at Harvard
Kanaka_Rajan
Lyman-break galaxy that is one of the oldest galaxies known
or a revolution?". Scientific American. Retrieved 16 September 2022. Kempner, Joshua (2022). "Cosmology Calculator". Retrieved 14 July 2024. Castellano
GHZ2
Class of cell surface receptors
PMID 28871306. Hanner M, Moebius FF, Flandorfer A, Knaus HG, Striessnig J, Kempner E, Glossmann H (July 1996). "Purification, molecular cloning, and expression
Sigma_receptor
Private university in Cambridge, Massachusetts, US
Letters Written at Bicentenary": "September 8th, 1936: As the first formal function in the celebration of Harvard's tercentenary, the Harvard Alumni Association
Harvard_University
Chaperone protein
S2CID 72726251. Hanner M, Moebius FF, Flandorfer A, Knaus HG, Striessnig J, Kempner E, et al. (July 1996). "Purification, molecular cloning, and expression
Sigma-1_receptor
Mathematical problem in number theory
(in German). 66 (1): 95–101. doi:10.1007/BF01450913. S2CID 121386035. Kempner, Aubrey (1912). "Bemerkungen zum Waringschen Problem". Mathematische Annalen
Waring's_problem
Post–World War II war crimes trials
International Military Tribunal for the Far East. Lewiston, New York: Robert M.W. Kempner Collegium by E. Edwin Mellen Press. ISBN 978-0-7734-8313-2. University
International Military Tribunal for the Far East
International_Military_Tribunal_for_the_Far_East
Law enforcement command structure
Quartermaster Section Deputy Commissioner, Public Information – Delaney Kempner Creative Services Section Digital Communications Section Public Information
Organization of the New York City Police Department
Organization_of_the_New_York_City_Police_Department
Self-enforced restraint from pleasurable activities
Mortification of the flesh Temperance Self-denial List of books about sobriety Kempner, Martha (2015). "Abstinence". The International Encyclopedia of Human Sexuality
Abstinence
German mathematician (1877–1938)
Number Theory, American Mathematical Society. ISBN 0-8218-2004-4. Landau's function Landau prime ideal theorem Landau's problems Landau's symbol (Big O notation)
Edmund_Landau
Alternative decimal expansion of 1
503; Bartle & Sherbert (1982), p. 61. Komornik & Loreti (1998), p. 636. Kempner (1936), p. 611; Petkovšek (1990), p. 409. Petkovšek (1990), pp. 410–411
0.999...
and asks to have him followed. After Lily's description, the boss, Mr. Kempner (Ken Lynch), decides to use Agent 702. Herman draws a sketch based on the
List_of_The_Munsters_episodes
Undergraduate math course at Harvard University
(2021) [1978]. Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable (3rd ed.). American Mathematical Society. ISBN 978-1-470-46767-8
Math_55
College in New London, Connecticut, US
National Baseball Hall of Fame director Jeff Idelson, philanthropist Nan Kempner, Beyond Meat founder Ethan Brown, Senior Federal District Judge Kimba Wood
Connecticut_College
City in Texas, United States
port-related industries. In 1905 William Lewis Moody, Jr. and Isaac H. Kempner, members of two of Galveston's leading families, founded the American National
Galveston,_Texas
Enzyme
doi:10.1016/j.tiv.2004.07.001. PMID 15826809. Saenz-Méndez P, Katz A, Pérez-Kempner ML, Ventura ON, Vázquez M (April 2017). "Structural insights into human
Microsomal_epoxide_hydrolase
German diplomat (1883-1968)
Verz. d. Gerichtspersonen u. Zeugen u. Einführungen, Ed. Robert M. W. Kempner and Carl Haensel with C. H. Tuerck, Schwäbisch Gmünd: Bürger, 1950, OCLC 603853607
Karl_Ritter_(diplomat)
School in Odessa, Texas, United States
Satin Strings is regularly asked to play at major state and national functions. In 1994, Satin Strings played at the D-Day celebrations in Normandy,
Permian_High_School
Non-standard positional numeral system
radix conversion. Negative bases were later mentioned in passing by A. J. Kempner in 1936 and studied in more detail by Zdzisław Pawlak and A. Wakulicz in
Negative_base
Fast-food restaurant in Atlanta, Georgia, U.S.
Archived from the original on January 31, 2016. Retrieved February 16, 2024. Kempner, Matt (February 11, 2016). "The Varsity: Owners of Atlanta icon face a
The_Varsity_(restaurant)
travel, tourism, insurance
KEMPNER FUNCTION
KEMPNER FUNCTION
Boy/Male
Gaelic
Brave chieftain.
Surname or Lastname
English
English : variant of Kilner.German, Dutch, and Jewish (Ashkenazic) : variant spelling of Kellner, in any of its senses: ‘cellarman’, ‘steward’, ‘overseer’, or ‘waiter’. In this spelling it is also found as a Czech name.Jewish (Ashkenazic) : occupational name from modern German Kellner or Yiddish kelner ‘waiter’.
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : habitational name from any of the various places in northern France called Saint-Pierre, from the dedication of their churches to St. Peter (see Peter).Eastern German : from a medieval personal name Sindperht, from sind ‘journey’ + berht ‘shining’.possibly variant of Catalan or Aragonese Samper.
Boy/Male
English
Keeper of horses.
Boy/Male
Tamil
Honey keeper
Surname or Lastname
English, Scottish, Dutch, and North German
English, Scottish, Dutch, and North German : variant of Kemp ‘champion’.Dutch : variant of Kempen 1.
Boy/Male
English
warrior.
Boy/Male
Sikh
Lotus keeper
Boy/Male
British, English
Park Keeper
Boy/Male
Muslim
Keeper. Guardian. Preserver.
Boy/Male
Irish
Cattle keeper.
Surname or Lastname
English
English : occupational name for a wool or flax comber, Middle English kem(be)stere (an agent derivative of Old English cemban ‘to comb’). Although this was originally a feminine form of the masculine kembere, by the Middle English period the suffix -stre had lost its feminine force, and the term was used to refer to both sexes. Compare Baxter, Brewster, Dexter.
Surname or Lastname
English
English : occupational name for a servant of the Knights Templar (see Temple).
Boy/Male
American, British, English
Warrior; Champion
Girl/Female
Biblical
Keeper, or keeping.
Boy/Male
Biblical
Keeper, dregs.
Girl/Female
American, Australian, British, Chinese, Christian, English
Park Keeper; Keeper of the Forest; Forest Ranger
Biblical
keeper; dregs
Boy/Male
Hindu
Honey keeper
Girl/Female
British, English
Keeper of Records
KEMPNER FUNCTION
KEMPNER FUNCTION
KEMPNER FUNCTION
KEMPNER FUNCTION
KEMPNER FUNCTION
KEMPNER FUNCTION
KEMPNER FUNCTION
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