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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.
In particular, can the nilpotency class of Q be higher than 3? Proposed: at Loops '07, Prague 2007; Comments: When the inner mapping group Inn(Q) is finite and abelian, then Q is nilpotent (Niemenaa and Kepka). The first question is therefore open only in the infinite case. Call loop Q of Csörgõ type if it is nilpotent of class at least 3 ...
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.
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 ...
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions.
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.
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For example, in most systems of logic (but not in intuitionistic logic) Peirce's law (((P→Q)→P)→P) is a theorem. For classical logic, it can be easily verified with a truth table . The study of mathematical proof is particularly important in logic, and has accumulated to automated theorem proving and formal verification of software.