# formal series

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##### 1: 16.11 Asymptotic Expansions
###### §16.11(i) FormalSeries
For subsequent use we define two formal infinite series, $E_{p,q}(z)$ and $H_{p,q}(z)$, as follows:
16.11.1 $E_{p,q}(z)=(2\pi)^{\ifrac{(p-q)}{2}}\kappa^{-\nu-(\ifrac{1}{2})}{\mathrm{e}}^{% \kappa z^{\ifrac{1}{\kappa}}}\sum_{k=0}^{\infty}c_{k}\left(\kappa z^{\ifrac{1}% {\kappa}}\right)^{\nu-k},$ $p,
16.11.2 $H_{p,q}(z)=\sum_{m=1}^{p}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\Gamma\left(a_{% m}+k\right)\left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}\ell=1\\ \ell\neq m\end{subarray}}^{p}\Gamma\left(a_{\ell}-a_{m}-k\right)}{\prod\limits% _{\ell=1}^{q}\Gamma\left(b_{\ell}-a_{m}-k\right)}}\right)z^{-a_{m}-k}.$
The formal series (16.11.2) for $H_{q+1,q}(z)$ converges if $\left|z\right|>1$, and …
##### 2: 10.70 Zeros
Let $\mu=4\nu^{2}$ and $f(t)$ denote the formal series
##### 3: 3.10 Continued Fractions
Every convergent, asymptotic, or formal seriesWe say that it corresponds to the formal power seriesWe say that it is associated with the formal power series $f(z)$ in (3.10.7) if the expansion of its $n$th convergent $C_{n}$ in ascending powers of $z$, agrees with (3.10.7) up to and including the term in $z^{2n-1}$, $n=1,2,3,\dots$. …
##### 4: 1.17 Integral and Series Representations of the Dirac Delta
By analogy with §1.17(ii) we have the formal series representation …
Formally, … …
##### 6: 16.5 Integral Representations and Integrals
In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as $z\to 0$ in the sector $|\operatorname{ph}\left(-z\right)|\leq(p+1-q-\delta)\pi/2$, where $\delta$ is an arbitrary small positive constant. …
##### 7: 2.1 Definitions and Elementary Properties
Let $\sum a_{s}x^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$, …
##### 8: Bibliography B
• L. J. Billera, C. Greene, R. Simion, and R. P. Stanley (Eds.) (1996) Formal Power Series and Algebraic Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, American Mathematical Society, Providence, RI.
• ##### 9: 16.4 Argument Unity
Contiguous balanced series have parameters shifted by an integer but still balanced. … … when the series on the right terminates and the series on the left converges. …
###### §16.4(v) Bilateral Series
Denote, formally, the bilateral hypergeometric function …
##### 10: 3.11 Approximation Techniques
be a formal power series. …