![gamma simbl gamma simbl](https://image.shutterstock.com/image-vector/gamma-greek-letter-icon-symbol-450w-450797314.jpg)
We think that it is simpler than to introduce a version of r + 1 V r with additive parameters. However, we decided to use the old Frenkel–Turaev notation for the reason that the additive parameters α j are more convenient for us than their exponentiated counterparts. We understand that the modern notation is better justified by the meaning of the elliptic very-well-poisedness condition than the old one and is really convenient in many cases. In the modern notation, what we call r + 1 ω r ( α 1 α 4, …, α r + 1 | η, τ ) (following ), would be r + 3 V r + 1 ( a 1 a 6, …, a r + 3 | q 2, p ) with q = e 2 π i η, p = e 2 π i τ, a j = e 4 π i η α j − 2. Which is valid provided that the balancing condition 2 α 1 + 1 = α 4 + α 5 + α 6 + α 7 − n is satisfied (the Frenkel–Turaev summation formula ).Ī remark on the notation is in order.
Gamma simbl series#
The logarithmic function −log(1 − z) has a hypergeometric series in the open unit disk, but it can be continued analytically to the complex plane with a cut along [1, ∞) and a branch point at z = 1. Take for example the geometric series, then it is clear that the hypergeometric series converges in the open unit disk, but the corresponding hypergeometric function is defined in the whole complex plane with a simple pole at z = 1. The analytic continuation of the hypergeometric series is then called the hypergeometric function. For p = q + 1, the hypergeometric series only converges in the open unit disk, but sometimes it can be continued analytically to a larger domain in the complex plane. For q ≥ p, the hypergeometric series therefore defines an entire function which is the corresponding hypergeometric function. None of the denominator parameters is allowed to be a negative integer − m, unless there is a numerator parameter which is a negative integer − k with k < m. When one of the numerator parameters is a negative integer, say a 1 = − m, then the series is terminating and defines a polynomial of degree m. ( z / 2 ) ν F 1 0 ( − ν + 1 − z 2 / 4 ) = Γ ( ν + 1 ) J ν ( z )įor generic values of the parameters, we see that the hypergeometric series converges everywhere in the complex plane when q ≥ p, it converges for | z | q + 1 it is only defined at z = 0.