Generalized Hypergeometric Functions & Meijer G -Function - 16.12 Products

From testwiki
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
16.12.E1 F 1 0 ⁡ ( - ; a ; z ) ⁢ F 1 0 ⁡ ( - ; b ; z ) = F 3 2 ⁡ ( 1 2 ⁢ ( a + b ) , 1 2 ⁢ ( a + b - 1 ) a , b , a + b - 1 ; 4 ⁢ z ) Gauss-hypergeometric-pFq 0 1 𝑎 𝑧 Gauss-hypergeometric-pFq 0 1 𝑏 𝑧 Gauss-hypergeometric-pFq 2 3 1 2 𝑎 𝑏 1 2 𝑎 𝑏 1 𝑎 𝑏 𝑎 𝑏 1 4 𝑧 {\displaystyle{\displaystyle{{}_{0}F_{1}}\left(-;a;z\right){{}_{0}F_{1}}\left(% -;b;z\right)={{}_{2}F_{3}}\left({\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)\atop a,b,% a+b-1};4z\right)}}
\genhyperF{0}{1}@{-}{a}{z}\genhyperF{0}{1}@{-}{b}{z} = \genhyperF{2}{3}@@{\frac{1}{2}(a+b),\frac{1}{2}(a+b-1)}{a,b,a+b-1}{4z}		 

hypergeom([-], [a], z)*hypergeom([-], [b], z) = hypergeom([(1)/(2)*(a + b),(1)/(2)*(a + b - 1)], [a , b , a + b - 1], 4*z)

HypergeometricPFQ[{-}, {a}, z]*HypergeometricPFQ[{-}, {b}, z] == HypergeometricPFQ[{Divide[1,2]*(a + b),Divide[1,2]*(a + b - 1)}, {a , b , a + b - 1}, 4*z]
Error Failure - Error
16.12.E2 ( F 1 2 ⁡ ( a , b a + b + 1 2 ; z ) ) 2 = F 2 3 ⁡ ( 2 ⁢ a , 2 ⁢ b , a + b a + b + 1 2 , 2 ⁢ a + 2 ⁢ b ; z ) superscript Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑎 𝑏 1 2 𝑧 2 Gauss-hypergeometric-pFq 3 2 2 𝑎 2 𝑏 𝑎 𝑏 𝑎 𝑏 1 2 2 𝑎 2 𝑏 𝑧 {\displaystyle{\displaystyle\left({{}_{2}F_{1}}\left({a,b\atop a+b+\frac{1}{2}% };z\right)\right)^{2}={{}_{3}F_{2}}\left({2a,2b,a+b\atop a+b+\frac{1}{2},2a+2b% };z\right)}}
\left(\genhyperF{2}{1}@@{a,b}{a+b+\frac{1}{2}}{z}\right)^{2} = \genhyperF{3}{2}@@{2a,2b,a+b}{a+b+\frac{1}{2},2a+2b}{z}		 

(hypergeom([a , b], [a + b +(1)/(2)], z))^(2) = hypergeom([2*a , 2*b , a + b], [a + b +(1)/(2), 2*a + 2*b], z)

(HypergeometricPFQ[{a , b}, {a + b +Divide[1,2]}, z])^(2) == HypergeometricPFQ[{2*a , 2*b , a + b}, {a + b +Divide[1,2], 2*a + 2*b}, z]
Failure Failure Manual Skip!
Failed [108 / 252]
Result: Complex[4.205771365940054, 0.2846096908265261]
Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-23.50000000000001, -28.578838324886455]
Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
16.12.E3 ( F 1 2 ⁡ ( a , b c ; z ) ) 2 = ∑ k = 0 ∞ ( 2 ⁢ a ) k ⁢ ( 2 ⁢ b ) k ⁢ ( c - 1 2 ) k ( c ) k ⁢ ( 2 ⁢ c - 1 ) k ⁢ k ! ⁢ F 3 4 ⁡ ( - 1 2 ⁢ k , 1 2 ⁢ ( 1 - k ) , a + b - c + 1 2 , 1 2 a + 1 2 , b + 1 2 , 3 2 - k - c ; 1 ) ⁢ z k superscript Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 2 superscript subscript 𝑘 0 Pochhammer 2 𝑎 𝑘 Pochhammer 2 𝑏 𝑘 Pochhammer 𝑐 1 2 𝑘 Pochhammer 𝑐 𝑘 Pochhammer 2 𝑐 1 𝑘 𝑘 Gauss-hypergeometric-pFq 4 3 1 2 𝑘 1 2 1 𝑘 𝑎 𝑏 𝑐 1 2 1 2 𝑎 1 2 𝑏 1 2 3 2 𝑘 𝑐 1 superscript 𝑧 𝑘 {\displaystyle{\displaystyle\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)% \right)^{2}=\sum_{k=0}^{\infty}\frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}% }{\left(c-\frac{1}{2}\right)_{k}}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}% k!}{{}_{4}F_{3}}\left({-\frac{1}{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{% 1}{2}\atop a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k}}}
\left(\genhyperF{2}{1}@@{a,b}{c}{z}\right)^{2} = \sum_{k=0}^{\infty}\frac{\Pochhammersym{2a}{k}\Pochhammersym{2b}{k}\Pochhammersym{c-\frac{1}{2}}{k}}{\Pochhammersym{c}{k}\Pochhammersym{2c-1}{k}k!}\genhyperF{4}{3}@@{-\frac{1}{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{1}{2}}{a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-c}{1}z^{k}		 | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}} 
(hypergeom([a , b], [c], z))^(2) = sum((pochhammer(2*a, k)*pochhammer(2*b, k)*pochhammer(c -(1)/(2), k))/(pochhammer(c, k)*pochhammer(2*c - 1, k)*factorial(k))*hypergeom([-(1)/(2)*k ,(1)/(2)*(1 - k), a + b - c +(1)/(2),(1)/(2)], [a +(1)/(2), b +(1)/(2),(3)/(2)- k - c], 1)*(z)^(k), k = 0..infinity)

(HypergeometricPFQ[{a , b}, {c}, z])^(2) == Sum[Divide[Pochhammer[2*a, k]*Pochhammer[2*b, k]*Pochhammer[c -Divide[1,2], k],Pochhammer[c, k]*Pochhammer[2*c - 1, k]*(k)!]*HypergeometricPFQ[{-Divide[1,2]*k ,Divide[1,2]*(1 - k), a + b - c +Divide[1,2],Divide[1,2]}, {a +Divide[1,2], b +Divide[1,2],Divide[3,2]- k - c}, 1]*(z)^(k), {k, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [159 / 216]
Result: -.1250000000
Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2}


Result: 5.872053804
Test Values: {a = -3/2, b = -3/2, c = -1/2, z = 1/2}

... skip entries to safe data
 Skipped - Because timed out
Retrieved from "https://lct.wmflabs.org/w/index.php?title=16.12&oldid=58318"