Interpolation on Gauss hypergeometric functions with an application.(English)Zbl 06864400

Summary: We use some standard numerical techniques to approximate the hypergeometric function ${}_2F_1[a,b;c;x]=1+\frac{ab}{c}x+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{x^2}{2!}+\cdots$ for a range of parameter triples $$(a,b,c)$$ on the interval $$0<x<1$$. Some of the familiar hypergeometric functional identities and asymptotic behavior of the hypergeometric function at $$x=1$$ play crucial roles in deriving the formula for such approximations. We also focus on error analysis of the numerical approximations leading to monotone properties of quotients of gamma functions in parameter triples $$(a,b,c)$$. Finally, an application to continued fractions of Gauss is discussed followed by concluding remarks consisting of recent works on related problems.

MSC:

 65D05 Numerical interpolation 33B15 Gamma, beta and polygamma functions 33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) 33C05 Classical hypergeometric functions, $${}_2F_1$$ 33F05 Numerical approximation and evaluation of special functions
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References:

 [1] 10.2307/2153171 · Zbl 0802.33001 [2] 10.1090/S0002-9939-97-04152-X · Zbl 0881.33001 [3] 10.1017/CBO9781107325937 [4] ; Bailey, Generalized hypergeometric series. Cambridge Tracts in Math. and Math. Phys., 32 (1935) · Zbl 0011.02303 [5] 10.1017/CBO9780511762543 [6] 10.2307/2008180 · Zbl 0607.33002 [7] 10.1016/j.disc.2011.04.019 · Zbl 1228.05060 [8] 10.1016/j.amc.2014.05.034 · Zbl 1334.33004 [9] 10.1002/sapm195938177 · Zbl 0094.04104 [10] 10.1016/S0377-0427(00)00659-2 · Zbl 0985.33003 [11] 10.1007/s11075-015-0070-y · Zbl 1382.33002 [12] 10.1007/s11139-016-9876-z · Zbl 1357.33004 [13] 10.1090/proc/13408 · Zbl 1371.33007 [14] 10.1007/s40314-015-0252-1 · Zbl 1359.33002 [15] 10.1007/s11075-016-0173-0 · Zbl 1360.33009
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