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Frank Hilton Jackson
The Reverend Frank Hilton Jackson (16 August 1870, Hull, England – 27 April 1960) was an English clergyman and mathematician who worked on basic hypergeometric series. He introduced several ''q''-analogs such as the Jackson–Bessel functions, the Jackson- Hahn-Cigler ''q''-addition, the Jackson derivative, and the Jackson integral In q-analog theory, the Jackson integral series in the theory of special functions that expresses the operation inverse to q-differentiation. The Jackson integral was introduced by Frank Hilton Jackson. For methods of numerical evaluation, see an .... Further reading *Ernst, T. (2012). A Comprehensive Treatment of q-Calculus. Springer Science & Business Media. *Gasper, G., Rahman, M.(2004). Basic Hypergeometric Series. Cambridge University Press. References * Selected papers * Jackson, F. H. (1917). The q-integral analogous to Borel's integral. Messenger Math, 47, 57–64. * Jackson, F. H. (1921). Summation of q-hypergeometric series. Messe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kingston Upon Hull
Kingston upon Hull, usually abbreviated to Hull, is a port city and unitary authority in the East Riding of Yorkshire, England. It lies upon the River Hull at its confluence with the Humber Estuary, inland from the North Sea and south-east of York, the historic county town. With a population of (), it is the fourth-largest city in the Yorkshire and the Humber region after Leeds, Sheffield and Bradford. The town of Wyke on Hull was founded late in the 12th century by the monks of Meaux Abbey as a port from which to export their wool. Renamed ''Kings-town upon Hull'' in 1299, Hull had been a market town, military supply port, trading centre, fishing and whaling centre and industrial metropolis. Hull was an early theatre of battle in the English Civil Wars. Its 18th-century Member of Parliament, William Wilberforce, took a prominent part in the abolition of the slave trade in Britain. More than 95% of the city was damaged or destroyed in the blitz and suffered a perio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
England
England is a country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest and the Celtic Sea to the southwest. It is separated from continental Europe by the North Sea to the east and the English Channel to the south. The country covers five-eighths of the island of Great Britain, which lies in the North Atlantic, and includes over 100 smaller islands, such as the Isles of Scilly and the Isle of Wight. The area now called England was first inhabited by modern humans during the Upper Paleolithic period, but takes its name from the Angles, a Germanic tribe deriving its name from the Anglia peninsula, who settled during the 5th and 6th centuries. England became a unified state in the 10th century and has had a significant cultural and legal impact on the wider world since the Age of Discovery, which began during the 15th century. The English language, the Anglican Church, and Engli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basic Hypergeometric Series
In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x''''n'' is called hypergeometric if the ratio of successive terms ''x''''n''+1/''x''''n'' is a rational function of ''n''. If the ratio of successive terms is a rational function of ''q''''n'', then the series is called a basic hypergeometric series. The number ''q'' is called the base. The basic hypergeometric series _2\phi_1(q^,q^;q^;q,x) was first considered by . It becomes the hypergeometric series F(\alpha,\beta;\gamma;x) in the limit when base q =1. Definition There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ. The unilateral basic hypergeometric series is defined as :\;_\phi_k \left begin a_1 & a_2 & \ldots & a_ \\ b_1 & b_2 & \ldots & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jackson–Bessel Function
In mathematics, a Jackson ''q''-Bessel function (or basic Bessel function) is one of the three ''q''-analogs of the Bessel function introduced by . The third Jackson ''q''-Bessel function is the same as the Hahn–Exton ''q''-Bessel function. Definition The three Jackson ''q''-Bessel functions are given in terms of the ''q''-Pochhammer symbol and the basic hypergeometric function \phi by : J_\nu^(x;q) = \frac (x/2)^\nu _2\phi_1(0,0;q^;q,-x^2/4), \quad , x, -1, the second Jackson ''q''-Bessel function satisfies: \left, J_^(z;q)\\leq\frac\left(\frac\right)^\nu\exp\left\. (see .) For n\in\mathbb, \left, J_^(z;q)\\leq\frac\left(\frac\right)^n(-, z, ^2;q)_. (see .) Generating Function The following formulas are the ''q''-analog of the generating function for the Bessel function (see ): :\sum_^t^nJ_n^(x;q)=(-x^2/4;q)_e_q(xt/2)e_q(-x/2t), :\sum_^t^nJ_n^(x;q)=e_q(xt/2)E_q(-qx/2t). e_q is the ''q''-exponential function. Alternative Representations Integral Representations The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfgang Hahn
Wolfgang Hahn (April 30, 1911 – January 10, 1998) was a German mathematician who worked on special functions, in particular orthogonal polynomials. He introduced Hahn polynomials, Hahn difference, Hahn q-addition (or Jackson-Hahn-Cigler q-addition), and the Hahn–Exton q-Bessel function. He was an honorary member of the Austrian Mathematical Society The Austrian Mathematical Society (german: Österreichische Mathematische Gesellschaft) is the national mathematical society of Austria and a member society of the European Mathematical Society. History The society was founded in 1903 by Ludwig B .... References * * * External links *Pictures of Wolfgang Hahn from Oberwolfach {{DEFAULTSORT:Hahn, Wolfgang 1911 births 1998 deaths Academic staff of the Technical University of Braunschweig 20th-century German mathematicians Q-analogs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jackson Derivative
In mathematics, in the area of combinatorics and quantum calculus, the ''q''-derivative, or Jackson derivative, is a ''q''-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's ''q''-integration. For other forms of q-derivative, see . Definition The ''q''-derivative of a function ''f''(''x'') is defined as :\left(\frac\right)_q f(x)=\frac. It is also often written as D_qf(x). The ''q''-derivative is also known as the Jackson derivative. Formally, in terms of Lagrange's shift operator in logarithmic variables, it amounts to the operator :D_q= \frac ~ \frac ~, which goes to the plain derivative \to \frac as q \to 1. It is manifestly linear, :\displaystyle D_q (f(x)+g(x)) = D_q f(x) + D_q g(x)~. It has a product rule analogous to the ordinary derivative product rule, with two equivalent forms :\displaystyle D_q (f(x)g(x)) = g(x)D_q f(x) + f(qx)D_q g(x) = g(qx)D_q f(x) + f(x)D_q g(x). Similarly, it satisfies a quotient rule, : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jackson Integral
In q-analog theory, the Jackson integral series in the theory of special functions that expresses the operation inverse to q-differentiation. The Jackson integral was introduced by Frank Hilton Jackson. For methods of numerical evaluation, see and . Definition Let ''f''(''x'') be a function of a real variable ''x''. For ''a'' a real variable, the Jackson integral of ''f'' is defined by the following series expansion: : \int_0^a f(x)\,_q x = (1-q)\,a\sum_^q^k f(q^k a). Consistent with this is the definition for a \to \infty \int_0^\infty f(x)\,_q x = (1-q)\sum_^q^k f(q^k ). More generally, if ''g''(''x'') is another function and ''D''''q''''g'' denotes its ''q''-derivative, we can formally write : \int f(x)\,D_q g\,_q x = (1-q)\,x\sum_^q^k f(q^k x)\,D_q g(q^k x) = (1-q)\,x\sum_^q^k f(q^k x)\tfrac, or : \int f(x)\,_q g(x) = \sum_^ f(q^k x)\cdot(g(q^x)-g(q^x)), giving a ''q''-analogue of the Riemann–Stieltjes integral. Jackson integral as q-antiderivative J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1870 Births
Year 187 ( CLXXXVII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Quintius and Aelianus (or, less frequently, year 940 '' Ab urbe condita''). The denomination 187 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Roman Empire * Septimius Severus marries Julia Domna (age 17), a Syrian princess, at Lugdunum (modern-day Lyon). She is the youngest daughter of high-priest Julius Bassianus – a descendant of the Royal House of Emesa. Her elder sister is Julia Maesa. * Clodius Albinus defeats the Chatti, a highly organized German tribe that controlled the area that includes the Black Forest. By topic Religion * Olympianus succeeds Pertinax as bishop of Byzantium (until 198). Births * Cao Pi, Chinese emperor of the Cao Wei state (d. 226) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1960 Deaths
Year 196 ( CXCVI) was a leap year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Dexter and Messalla (or, less frequently, year 949 ''Ab urbe condita''). The denomination 196 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Roman Empire * Emperor Septimius Severus attempts to assassinate Clodius Albinus but fails, causing Albinus to retaliate militarily. * Emperor Septimius Severus captures and sacks Byzantium; the city is rebuilt and regains its previous prosperity. * In order to assure the support of the Roman legion in Germany on his march to Rome, Clodius Albinus is declared Augustus by his army while crossing Gaul. * Hadrian's wall in Britain is partially destroyed. China * First year of the '' Jian'an era of the Chinese Han Dynasty. * Emperor Xian o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
English Mathematicians
English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national identity, an identity and common culture ** English language in England, a variant of the English language spoken in England * English languages (other) * English studies, the study of English language and literature * ''English'', an Amish term for non-Amish, regardless of ethnicity Individuals * English (surname), a list of notable people with the surname ''English'' * People with the given name ** English McConnell (1882–1928), Irish footballer ** English Fisher (1928–2011), American boxing coach ** English Gardner (b. 1992), American track and field sprinter Places United States * English, Indiana, a town * English, Kentucky, an unincorporated community * English, Brazoria County, Texas, an unincorporated community * Engli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-analogs
In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q''-analogs that arise naturally, rather than in arbitrarily contriving ''q''-analogs of known results. The earliest ''q''-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, , , ''q''-analogues are most frequently studied in the mathematical fields of combinatorics and special functions. In these settings, the limit is often formal, as is often discrete-valued (for example, it may represent a prime power). ''q''-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of chaotic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
