# large variable

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##### 7: 16.11 Asymptotic Expansions
###### §16.11(ii) Expansions for LargeVariable
16.11.10 ${{}_{p+1}F_{p}}\left({a_{1}+r,\dots,a_{k-1}+r,a_{k},\dots,a_{p+1}\atop b_{1}+r% ,\dots,b_{k}+r,b_{k+1},\dots,b_{p}};z\right)=\sum_{n=0}^{m-1}\frac{{\left(a_{1% }+r\right)_{n}}\cdots{\left(a_{k-1}+r\right)_{n}}{\left(a_{k}\right)_{n}}% \cdots{\left(a_{p+1}\right)_{n}}}{{\left(b_{1}+r\right)_{n}}\cdots{\left(b_{k}% +r\right)_{n}}{\left(b_{k+1}\right)_{n}}\cdots{\left(b_{p}\right)_{n}}}\frac{z% ^{n}}{n!}+O\left(\frac{1}{r^{m}}\right),$
16.11.11 ${{}_{p}F_{q}}\left({a_{1}+r,\dots,a_{p}+r\atop b_{1}+r,\dots,b_{q}+r};z\right)% =\sum_{n=0}^{m-1}\frac{{\left(a_{1}+r\right)_{n}}\cdots{\left(a_{p}+r\right)_{% n}}}{{\left(b_{1}+r\right)_{n}}\cdots{\left(b_{q}+r\right)_{n}}}\frac{z^{n}}{n% !}+O\left(\frac{1}{r^{(q-p)m}}\right),$
##### 9: Bibliography F
• C. Ferreira, J. L. López, and E. Pérez Sinusía (2013a) The third Appell function for one large variable. J. Approx. Theory 165, pp. 60–69.
• C. Ferreira, J. L. López, and E. Pérez Sinusía (2005) Incomplete gamma functions for large values of their variables. Adv. in Appl. Math. 34 (3), pp. 467–485.
• C. Ferreira, J. L. López, and E. P. Sinusía (2013b) The second Appell function for one large variable. Mediterr. J. Math. 10 (4), pp. 1853–1865.