# rational functions

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##### 1: 16.26 Approximations
For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).
##### 2: 5.19 Mathematical Applications
###### §5.19(i) Summation of RationalFunctions
As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions. …
##### 4: 5.23 Approximations
###### §5.23(iii) Approximations in the Complex Plane
See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of $\Gamma\left(z\right)$. …
##### 5: 16.7 Relations to Other Functions
Further representations of special functions in terms of ${{}_{p}F_{q}}$ functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of ${{}_{q+1}F_{q}}$ functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).
##### 6: 19.2 Definitions
Let $s^{2}(t)$ be a cubic or quartic polynomial in $t$ with simple zeros, and let $r(s,t)$ be a rational function of $s$ and $t$ containing at least one odd power of $s$. …
19.2.2 $r(s,t)=\frac{(p_{1}+p_{2}s)(p_{3}-p_{4}s)s}{(p_{3}+p_{4}s)(p_{3}-p_{4}s)s}=% \frac{\rho}{s}+\sigma,$
where $p_{j}$ is a polynomial in $t$ while $\rho$ and $\sigma$ are rational functions of $t$. …
##### 7: 3.11 Approximation Techniques
###### §3.11(iii) Minimax Rational Approximations
The theory of polynomial minimax approximation given in §3.11(i) can be extended to the case when $p_{n}(x)$ is replaced by a rational function $R_{k,\ell}(x)$. …
###### Example
The rational functionA general procedure is to approximate $F$ by a rational function $R$ (vanishing at infinity) and then approximate $f$ by $r={\mathscr{L}}^{-1}R$. …
##### 8: Bibliography I
• M. E. H. Ismail and D. R. Masson (1994) $q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.
• ##### 9: 17.12 Bailey Pairs
A sequence of pairs of rational functions of several variables $(\alpha_{n},\beta_{n})$, $n=0,1,2,\dots$, is called a Bailey pair provided that for each $n\geqq 0$
##### 10: 31.8 Solutions via Quadratures
By automorphisms from §31.2(v), similar solutions also exist for $m_{0},m_{1},m_{2},m_{3}\in\mathbb{Z}$, and $\Psi_{g,N}(\lambda,z)$ may become a rational function in $z$. …