Parabolic Cylinder Functions - 12.13 Sums

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
12.13.E1 U ⁡ ( a , x + y ) = e 1 2 ⁢ x ⁢ y + 1 4 ⁢ y 2 ⁢ ∑ m = 0 ∞ ( - y ) m m ! ⁢ U ⁡ ( a - m , x ) parabolic-U 𝑎 𝑥 𝑦 superscript 𝑒 1 2 𝑥 𝑦 1 4 superscript 𝑦 2 superscript subscript 𝑚 0 superscript 𝑦 𝑚 𝑚 parabolic-U 𝑎 𝑚 𝑥 {\displaystyle{\displaystyle U\left(a,x+y\right)=e^{\frac{1}{2}xy+\frac{1}{4}y% ^{2}}\sum_{m=0}^{\infty}\frac{(-y)^{m}}{m!}U\left(a-m,x\right)}}
\paraU@{a}{x+y} = e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac{(-y)^{m}}{m!}\paraU@{a-m}{x}		 

CylinderU(a, x + y) = exp((1)/(2)*x*y +(1)/(4)*(y)^(2))*sum(((- y)^(m))/(factorial(m))*CylinderU(a - m, x), m = 0..infinity)

ParabolicCylinderD[- 1/2 -(a), x + y] == Exp[Divide[1,2]*x*y +Divide[1,4]*(y)^(2)]*Sum[Divide[(- y)^(m),(m)!]*ParabolicCylinderD[- 1/2 -(a - m), x], {m, 0, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
12.13.E2 U ⁡ ( a , x + y ) = e - 1 2 ⁢ x ⁢ y - 1 4 ⁢ y 2 ⁢ ∑ m = 0 ∞ ( - a - 1 2 m ) ⁢ y m ⁢ U ⁡ ( a + m , x ) parabolic-U 𝑎 𝑥 𝑦 superscript 𝑒 1 2 𝑥 𝑦 1 4 superscript 𝑦 2 superscript subscript 𝑚 0 binomial 𝑎 1 2 𝑚 superscript 𝑦 𝑚 parabolic-U 𝑎 𝑚 𝑥 {\displaystyle{\displaystyle U\left(a,x+y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}% y^{2}}\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt}{}{-a-\tfrac{1}{2}}{m}y^{m}U% \left(a+m,x\right)}}
\paraU@{a}{x+y} = e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\binom{-a-\tfrac{1}{2}}{m}y^{m}\paraU@{a+m}{x}		 

CylinderU(a, x + y) = exp(-(1)/(2)*x*y -(1)/(4)*(y)^(2))*sum(binomial(- a -(1)/(2),m)*(y)^(m)* CylinderU(a + m, x), m = 0..infinity)

ParabolicCylinderD[- 1/2 -(a), x + y] == Exp[-Divide[1,2]*x*y -Divide[1,4]*(y)^(2)]*Sum[Binomial[- a -Divide[1,2],m]*(y)^(m)* ParabolicCylinderD[- 1/2 -(a + m), x], {m, 0, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
12.13.E3 V ⁡ ( a , x + y ) = e 1 2 ⁢ x ⁢ y + 1 4 ⁢ y 2 ⁢ ∑ m = 0 ∞ ( a - 1 2 m ) ⁢ y m ⁢ V ⁡ ( a - m , x ) parabolic-V 𝑎 𝑥 𝑦 superscript 𝑒 1 2 𝑥 𝑦 1 4 superscript 𝑦 2 superscript subscript 𝑚 0 binomial 𝑎 1 2 𝑚 superscript 𝑦 𝑚 parabolic-V 𝑎 𝑚 𝑥 {\displaystyle{\displaystyle V\left(a,x+y\right)=e^{\frac{1}{2}xy+\frac{1}{4}y% ^{2}}\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt}{}{a-\tfrac{1}{2}}{m}y^{m}V\left(% a-m,x\right)}}
\paraV@{a}{x+y} = e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\binom{a-\tfrac{1}{2}}{m}y^{m}\paraV@{a-m}{x}		 

CylinderV(a, x + y) = exp((1)/(2)*x*y +(1)/(4)*(y)^(2))*sum(binomial(a -(1)/(2),m)*(y)^(m)* CylinderV(a - m, x), m = 0..infinity)

Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, x + y] + ParabolicCylinderD[-(a) - 1/2, -(x + y)]) == Exp[Divide[1,2]*x*y +Divide[1,4]*(y)^(2)]*Sum[Binomial[a -Divide[1,2],m]*(y)^(m)* Divide[GAMMA[1/2 + a - m], Pi]*(Sin[Pi*(a - m)] * ParabolicCylinderD[-(a - m) - 1/2, x] + ParabolicCylinderD[-(a - m) - 1/2, -(x)]), {m, 0, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
12.13.E4 V ⁡ ( a , x + y ) = e - 1 2 ⁢ x ⁢ y - 1 4 ⁢ y 2 ⁢ ∑ m = 0 ∞ y m m ! ⁢ V ⁡ ( a + m , x ) parabolic-V 𝑎 𝑥 𝑦 superscript 𝑒 1 2 𝑥 𝑦 1 4 superscript 𝑦 2 superscript subscript 𝑚 0 superscript 𝑦 𝑚 𝑚 parabolic-V 𝑎 𝑚 𝑥 {\displaystyle{\displaystyle V\left(a,x+y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}% y^{2}}\sum_{m=0}^{\infty}\frac{y^{m}}{m!}V\left(a+m,x\right)}}
\paraV@{a}{x+y} = e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac{y^{m}}{m!}\paraV@{a+m}{x}		 

CylinderV(a, x + y) = exp(-(1)/(2)*x*y -(1)/(4)*(y)^(2))*sum(((y)^(m))/(factorial(m))*CylinderV(a + m, x), m = 0..infinity)

Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, x + y] + ParabolicCylinderD[-(a) - 1/2, -(x + y)]) == Exp[-Divide[1,2]*x*y -Divide[1,4]*(y)^(2)]*Sum[Divide[(y)^(m),(m)!]*Divide[GAMMA[1/2 + a + m], Pi]*(Sin[Pi*(a + m)] * ParabolicCylinderD[-(a + m) - 1/2, x] + ParabolicCylinderD[-(a + m) - 1/2, -(x)]), {m, 0, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
12.13.E5 U ⁡ ( a , x ⁢ cos ⁡ t + y ⁢ sin ⁡ t ) = e 1 4 ⁢ ( x ⁢ sin ⁡ t - y ⁢ cos ⁡ t ) 2 ⁢ ∑ m = 0 ∞ ( - a - 1 2 m ) ⁢ ( tan ⁡ t ) m ⁢ U ⁡ ( m + a , x ) ⁢ U ⁡ ( - m - 1 2 , y ) parabolic-U 𝑎 𝑥 𝑡 𝑦 𝑡 superscript 𝑒 1 4 superscript 𝑥 𝑡 𝑦 𝑡 2 superscript subscript 𝑚 0 binomial 𝑎 1 2 𝑚 superscript 𝑡 𝑚 parabolic-U 𝑚 𝑎 𝑥 parabolic-U 𝑚 1 2 𝑦 {\displaystyle{\displaystyle U\left(a,x\cos t+y\sin t\right)\\ =e^{\frac{1}{4}(x\sin t-y\cos t)^{2}}\*\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt% }{}{-a-\tfrac{1}{2}}{m}(\tan t)^{m}U\left(m+a,x\right)U\left(-m-\tfrac{1}{2},y% \right)}}
\paraU@{a}{x\cos@@{t}+y\sin@@{t}}\\ = e^{\frac{1}{4}(x\sin@@{t}-y\cos@@{t})^{2}}\*\sum_{m=0}^{\infty}\binom{-a-\tfrac{1}{2}}{m}(\tan@@{t})^{m}\paraU@{m+a}{x}\paraU@{-m-\tfrac{1}{2}}{y}		 ℜ ⁡ a ≤ - 1 2 , 0 ≤ t , t ≤ 1 4 ⁢ π formulae-sequence 𝑎 1 2 formulae-sequence 0 𝑡 𝑡 1 4 𝜋 {\displaystyle{\displaystyle\Re a\leq-\tfrac{1}{2},0\leq t,t\leq\tfrac{1}{4}% \pi}} 
CylinderU(a, x*cos(t)+ y*sin(t)) = exp((1)/(4)*(x*sin(t)- y*cos(t))^(2))* sum(binomial(- a -(1)/(2),m)*(tan(t))^(m)* CylinderU(m + a, x)*CylinderU(- m -(1)/(2), y), m = 0..infinity)

ParabolicCylinderD[- 1/2 -(a), x*Cos[t]+ y*Sin[t]] == Exp[Divide[1,4]*(x*Sin[t]- y*Cos[t])^(2)]* Sum[Binomial[- a -Divide[1,2],m]*(Tan[t])^(m)* ParabolicCylinderD[- 1/2 -(m + a), x]*ParabolicCylinderD[- 1/2 -(- m -Divide[1,2]), y], {m, 0, Infinity}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skip - No test values generated
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