Bell series

In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell.

Given an arithmetic function and a prime , define the formal power series , called the Bell series of modulo as:

Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the uniqueness theorem: given multiplicative functions and , one has if and only if:

for all primes .

Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions and , let be their Dirichlet convolution. Then for every prime , one has:

In particular, this makes it trivial to find the Bell series of a Dirichlet inverse.

If is completely multiplicative, then formally:


The following is a table of the Bell series of well-known arithmetic functions.

  • The Möbius function has
  • The Mobius function squared has
  • Euler's totient has
  • The multiplicative identity of the Dirichlet convolution has
  • The Liouville function has
  • The power function Idk has Here, Idk is the completely multiplicative function .
  • The divisor function has
  • The unit function satisfies , i.e., is the geometric series.
  • If is the power of the prime omega function, then
  • Suppose that f is multiplicative and g is any arithmetic function satisfying for all primes p and . Then
  • If denotes the Mobius function of order k, then

See also


  • Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
1888 Miami Redskins football team

The 1888 Miami Redskins football team was an American football team that represented Miami University during the 1888 college football season. The 1888 team was Miami's first football team to compete in intercollegiate football. The team played only one game, a scoreless tie with the University of Cincinnati football team at Oxford, Ohio, on December 8, 1888. The team did not have a paid coach from 1888 to 1894. The 1888 game between Cincinnati and Miami was the first in what later became the Victory Bell series that has been included more than 110 games and is one of the oldest rivalries in college football. It was also the first college football game played in the State of Ohio.The players on the first Miami football team included W. Chidlaw, S. Fox, J. Lough, F. McCracken, J. Macready, R. Mason, E. Reed, S. Stephenson, W. Stubbs, and S. Townsend.

Eric Temple Bell

Eric Temple Bell (February 7, 1883 – December 21, 1960) was a Scottish-born mathematician and science fiction writer who lived in the United States for most of his life. He published non-fiction using his given name and fiction as John Taine.

Generating function

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a power series. This formal power series is the generating function. Unlike an ordinary series, this formal series is allowed to diverge, meaning that the generating function is not always a true function and the "variable" is actually 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 series in more than one indeterminate, to encode information about arrays of numbers indexed by several natural 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 that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.

Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations defined for formal series. These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of x. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal series as its series expansion; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a convergent series when a nonzero numeric value is substituted for x. Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series; for example, negative and fractional powers of x are examples of functions that do not have a corresponding formal power series.

Generating functions are not functions in the formal sense of a mapping from a domain to a codomain. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence of term coefficients.

Greg Rucka

Gregory Rucka (born November 29, 1969) is an American comic book writer and novelist, known for his work on such comics as Action Comics, Batwoman, Detective Comics, and the miniseries Superman: World of New Krypton for DC Comics, and for novels such as his Atticus Kodiak series.

Ikue Ōtani

Ikue Ōtani (大谷 育江, Ōtani Ikue, born August 18, 1965) is a Japanese actress, voice actress and narrator from Tokyo. She is best known for her anime roles in the Pokémon series (as Pikachu), One Piece (as Tony Tony Chopper), Naruto (as Konohamaru Sarutobi), Detective Conan (as Mitsuhiko Tsuburaya), Uchi no Sanshimai (as Fu), and Konjiki no Gash Bell (as Gash). She is currently attached to Mausu Promotion. Her pet name is "Iku-chan". She is known for playing both male and female roles, and sometimes plays multiple roles in one production. She is a native of Tokyo, but grew up in Niigata Prefecture.

Jenn Bennett

Jenn Bennett is an American author of novels for teens and adults. Her notable works include Alex, Approximately, Starry Eyes, and The Anatomical Shape of a Heart (aka Night Owls). Her books have received critical acclaim and award recognition.

Korean horror

Korean horror (K-horror) films have been around since the early years of Korean cinema, however, it was not until the late 1990s that the genre began to experience a renewal. Many of the Korean horror films tend to focus on the suffering and the anguish of characters rather than focus on the explicit "blood and guts" aspect of horror. Korean horror features many of the same motifs, themes, and imagery as Japanese horror.

Modern South Korean horror films are typically distinguished by stylish directing, themes of social commentary, and genre blending . The horror and thriller genres are cited as gaining international attention to South Korean Cinema.

American Hollywood has adapted several Korean horror films such as Oldboy (2003 film), Into the Mirror (2003), and A Tale of Two Sisters (2003). Train to Busan (2016) and The Wailing (film) (2016) are rumored to currently have remakes in talks for production.

Odo Hirsch

Odo Hirsch (born 1962) is the pen name of David Kausman, an Australian author of children's books. He was born in Melbourne, where he trained to be a doctor, but moved to London, where he currently lives.

After working as a doctor in both Melbourne and London, he joined Amnesty International, where he reported on torture victims and examined hospital conditions in Eastern Europe. After doing a master's degree in political thought at Cambridge University, he joined McKinsey Consulting in 1997. This was when his first novel for children, Antonio S and the Mystery of Theodore Guzman, was published. His other books include Bartlett and the Ice Voyage, which won the Blue Peter Book Award. His novels have been shortlisted many times for the Children's Book Council of Australia Awards.

Pietje Bell

The Pietje Bell series of novels (orig. Dutch: Pietje Bell boeken) is a series of children's books written by the Dutch writer Chris van Abkoude. It's about a little boy who gets himself into trouble.The story has been adapted to film twice. Once in 1964, as De Avonturen van Pietje Bell, by Henk van der Linden and second in 2002 and 2003 by Maria Peters.

In 2003 Ruud Bos adapted the novel into a musical called Pietje Bell - De Musical.

Dick Matena also adapted the novel into a comic book album.

Pinkerton (detective agency)

Pinkerton, founded as the Pinkerton National Detective Agency, is a private security guard and detective agency established in the United States by Scotsman Allan Pinkerton in 1850 and currently a subsidiary of Securitas AB. Pinkerton became famous when he claimed to have foiled a plot to assassinate president-elect Abraham Lincoln, who later hired Pinkerton agents for his personal security during the Civil War. Pinkerton's agents performed services ranging from security guarding to private military contracting work. Pinkerton was the largest private law enforcement organization in the world at the height of its power.During the labor strikes of the late 19th and early 20th centuries, businessmen hired the Pinkerton Agency to infiltrate unions, supply guards, keep strikers and suspected unionists out of factories, and recruit goon squads to intimidate workers. One such confrontation was the Homestead Strike of 1892, in which Pinkerton agents were called in to reinforce the strikebreaking measures of industrialist Henry Clay Frick, acting on behalf of Andrew Carnegie. The ensuing battle between Pinkerton agents and striking workers led to the deaths of seven Pinkerton agents and nine steelworkers. The Pinkertons were also used as guards in coal, iron, and lumber disputes in Illinois, Michigan, New York, Pennsylvania, and West Virginia as well as the Great Railroad Strike of 1877 and the Battle of Blair Mountain in 1921.

The company now operates as "Pinkerton Consulting & Investigations, Inc. d.b.a. Pinkerton Corporate Risk Management", a division of the Swedish security company Securitas AB. The former Government Services division, PGS, now operates as Securitas Critical Infrastructure Services, Inc.

Pixie Hollow Games

Pixie Hollow Games is a 30-minute television special broadcast on November 19, 2011, on Disney Channel. Based on the Disney Fairies franchise, it was produced by DisneyToon Studios and animated by Prana Studios. It features the voices of Mae Whitman, Lucy Liu, Raven-Symoné, Megan Hilty, Angela Bartys, and others, as Tinker Bell and the other fairies of Pixie Hollow in Never Land, taking part in an Olympic-style competition. It is based loosely on J. M. Barrie's Peter Pan stories, by way of Disney's animated adaptation.

It was originally intended as the last of five feature-length films in the Tinker Bell series of direct-to-DVD 3D animated films, with the title Tinker Bell: Race through the Seasons, and a release date in 2012. However, the movie was rescheduled and retooled as a TV special instead of a film-length movie. Unlike the previous feature films in this series, Tinker Bell is not a central featured character in this special release.

The Bell System Science Series

The Bell System Science Series consists of nine television specials made for the AT&T Corporation that were originally broadcast in color between 1956 and 1964. Marcel LaFollette has described them as "specials that combined clever story lines, sophisticated animation, veteran character actors, films of natural phenomena, interviews with scientists, and precise explanation of scientific and technical concepts — all in the pursuit of better public understanding of science." Geoff Alexander and Rick Prelinger have described the films as "among the best known and remembered educational films ever made, and enthroning Dr. Frank Baxter, professor at the University of Southern California, as something of a legend as the omniscient king of academic science films hosts."AT&T and its subsidiary Bell Telephone System had a history of sponsoring broadcasting such as the Bell Telephone Hour, which was a weekly radio program of classical and Broadway music. AT&T's advertising agency, N. W. Ayer & Son, suggested that they also sponsor "television specials aimed at family audiences. Science was a natural topic choice, given the accomplishments and reputation of the company’s research arm, Bell Telephone Laboratories." They ultimately approached the famed filmmaker Frank Capra, who had numerous nominations and wins for the Academy Award for Best Director in the 1930s and 40s for films such as It's a Wonderful Life (1946). Capra produced the four films that were televised from 1956–1958. The second four films were produced by Warner Bros. Pictures, with veteran filmmaker Owen Crump in charge; these were televised between 1958 and 1962. The final film was produced by Walt Disney Pictures, and televised in 1964. Each special explored a single subject in detail. The host for the first eight films was Dr. Frank C. Baxter, a USC professor of English and television personality who played the role of "Dr. Research" (or "Dr. Linguistics" in The Alphabet Conspiracy). The host for the last film in the series was Walt Disney.

Following their television broadcast, the films were made available at minimal cost for classroom use. J. B. Gilbert estimated that, by the mid 1960s, the films had been watched by 5 million schoolchildren and half a million college students"; about 1600 copies of the film were ultimately distributed. Most of the films have been released for home video.

LaFollette notes that, "Production approaches that are now standard practice on NOVA and the Discovery Channel derive, in fact, from experimentation by television pioneers like Lynn Poole and Don Herbert and such programs as Adventure, Zoo Parade, Science in Action, and the Bell Telephone System’s science specials. These early efforts were also influenced by television’s love of the dramatic, refined during its first decade and continuing to shape news and public affairs programming, as well as fiction and fantasy, today."

The Pirate Fairy

The Pirate Fairy (alternatively known as Tinker Bell and the Pirate Fairy) is a 2014 American 3D computer-animated fantasy film directed by Peggy Holmes. It is the fifth direct-to-video feature-length animated film in the DisneyToon Studios' Tinker Bell film series and the Disney Fairies franchise, based on the character Tinker Bell from J. M. Barrie's Peter and Wendy. The film features the voices of Mae Whitman, reprising her role of Tinker Bell, Christina Hendricks as a dust-keeper fairy named Zarina, and Tom Hiddleston as a much younger James Hook.

Tiffani Thiessen

Tiffani Amber Thiessen (born January 23, 1974) is an American actress. She is known for starring as Kelly Kapowski on NBC's Saved by the Bell (1989–93) and as Valerie Malone on Fox's Beverly Hills, 90210 (1994–98). Thiessen has also starred in other TV series such as Fox's Fastlane (2002–03), ABC's What About Brian (2007), and USA Network's White Collar (2009–14), as well as in a number of TV movies, and she has also appeared in several films like Son in Law (1993), Shriek If You Know What I Did Last Friday the Thirteenth (2000), Hollywood Ending (2002), and Cyborg Soldier (2008).

Torsten (Andy Bell series)

Torsten is a series of projects by pop singer Andy Bell of the band Erasure consisting of various stage acts performed by him, and accompanying soundtracks, concept albums and remix albums. The series was written by playwright Barney Ashton-Bullock, and music by Christopher Frost. Two major works emerged, Torsten the Bareback Saint (2014) and Torsten The Beautiful Libertine (2016) with two accompanying album / soundtracks and two Variance Remix albums.

United Shades of America

United Shades of America is a CNN American documentary television series starring W. Kamau Bell.'United Shades of America' follows comedian and political provocateur W. Kamau Bell as he explores communities across America to understand the unique challenges they face.

The show airs on CNN Sundays at 10:00 PM ET/PT, and won the Emmy Award for Outstanding Unstructured Reality Program category at the 69th Primetime Emmy Awards and the 70th Primetime Emmy Awards.

CNN renewed the series. A second season with eight new episodes aired in 2017. The third season began on April 29, 2018.

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