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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.
The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century. [1] q-analogs are most frequently studied in the mathematical fields of combinatorics and special functions. In these settings, the limit q → 1 is often formal, as q is often discrete-valued (for example, it may represent a ...
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.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
If p < q + 1 then the ratio of coefficients tends to zero. This implies that the series converges for any finite value of z and thus defines an entire function of z. An example is the power series for the exponential function. If p = q + 1 then the ratio of coefficients tends to one.
q-Bessel polynomials; q-Charlier polynomials; q-Hahn polynomials; q-Jacobi polynomials: Big q-Jacobi polynomials; Continuous q-Jacobi polynomials; Little q-Jacobi polynomials; q-Krawtchouk polynomials; q-Laguerre polynomials; q-Meixner polynomials; q-Meixner–Pollaczek polynomials; q-Racah polynomials
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The q-Pochhammer symbol is a major building block in the construction of q-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series.