Heun Functions - 31.11 Expansions in Series of Hypergeometric Functions
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 31.11.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle L_{0}c_{0}+M_{0}c_{1} = 0}
L_{0}c_{0}+M_{0}c_{1} = 0 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | L[0]*c[0]+ M[0]*c[1] = 0 |
Subscript[L, 0]*Subscript[c, 0]+ Subscript[M, 0]*Subscript[c, 1] == 0 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle K_{j}c_{j-1}+L_{j}c_{j}+M_{j}c_{j+1} = 0}
K_{j}c_{j-1}+L_{j}c_{j}+M_{j}c_{j+1} = 0 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | K[j]*c[j - 1]+ L[j]*c[j]+ M[j]*c[j + 1] = 0 |
Subscript[K, j]*Subscript[c, j - 1]+ Subscript[L, j]*Subscript[c, j]+ Subscript[M, j]*Subscript[c, j + 1] == 0 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle K_{j} = -\frac{(j+\alpha-\mu-1)(j+\beta-\mu-1)(j+\gamma-\mu-1)(j+\lambda-1)}{(2j+\lambda-\mu-1)(2j+\lambda-\mu-2)}}
K_{j} = -\frac{(j+\alpha-\mu-1)(j+\beta-\mu-1)(j+\gamma-\mu-1)(j+\lambda-1)}{(2j+\lambda-\mu-1)(2j+\lambda-\mu-2)} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | K[j] = -((j + alpha - mu - 1)*(j + beta - mu - 1)*(j + gamma - mu - 1)*(j + lambda - 1))/((2*j + lambda - mu - 1)*(2*j + lambda - mu - 2)) |
Subscript[K, j] == -Divide[(j + \[Alpha]- \[Mu]- 1)*(j + \[Beta]- \[Mu]- 1)*(j + \[Gamma]- \[Mu]- 1)*(j + \[Lambda]- 1),(2*j + \[Lambda]- \[Mu]- 1)*(2*j + \[Lambda]- \[Mu]- 2)] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle L_{j} = a(\lambda+j)(\mu-j)-q+\frac{(j+\alpha-\mu)(j+\beta-\mu)(j+\gamma-\mu)(j+\lambda)}{(2j+\lambda-\mu)(2j+\lambda-\mu+1)}+\frac{(j-\alpha+\lambda)(j-\beta+\lambda)(j-\gamma+\lambda)(j-\mu)}{(2j+\lambda-\mu)(2j+\lambda-\mu-1)}}
L_{j} = a(\lambda+j)(\mu-j)-q+\frac{(j+\alpha-\mu)(j+\beta-\mu)(j+\gamma-\mu)(j+\lambda)}{(2j+\lambda-\mu)(2j+\lambda-\mu+1)}+\frac{(j-\alpha+\lambda)(j-\beta+\lambda)(j-\gamma+\lambda)(j-\mu)}{(2j+\lambda-\mu)(2j+\lambda-\mu-1)} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | L[j] = a*(lambda + j)*(mu - j)- q +((j + alpha - mu)*(j + beta - mu)*(j + gamma - mu)*(j + lambda))/((2*j + lambda - mu)*(2*j + lambda - mu + 1))+((j - alpha + lambda)*(j - beta + lambda)*(j - gamma + lambda)*(j - mu))/((2*j + lambda - mu)*(2*j + lambda - mu - 1)) |
Subscript[L, j] == a*(\[Lambda]+ j)*(\[Mu]- j)- q +Divide[(j + \[Alpha]- \[Mu])*(j + \[Beta]- \[Mu])*(j + \[Gamma]- \[Mu])*(j + \[Lambda]),(2*j + \[Lambda]- \[Mu])*(2*j + \[Lambda]- \[Mu]+ 1)]+Divide[(j - \[Alpha]+ \[Lambda])*(j - \[Beta]+ \[Lambda])*(j - \[Gamma]+ \[Lambda])*(j - \[Mu]),(2*j + \[Lambda]- \[Mu])*(2*j + \[Lambda]- \[Mu]- 1)] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle M_{j} = -\frac{(j-\alpha+\lambda+1)(j-\beta+\lambda+1)(j-\gamma+\lambda+1)(j-\mu+1)}{(2j+\lambda-\mu+1)(2j+\lambda-\mu+2)}}
M_{j} = -\frac{(j-\alpha+\lambda+1)(j-\beta+\lambda+1)(j-\gamma+\lambda+1)(j-\mu+1)}{(2j+\lambda-\mu+1)(2j+\lambda-\mu+2)} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | M[j] = -((j - alpha + lambda + 1)*(j - beta + lambda + 1)*(j - gamma + lambda + 1)*(j - mu + 1))/((2*j + lambda - mu + 1)*(2*j + lambda - mu + 2)) |
Subscript[M, j] == -Divide[(j - \[Alpha]+ \[Lambda]+ 1)*(j - \[Beta]+ \[Lambda]+ 1)*(j - \[Gamma]+ \[Lambda]+ 1)*(j - \[Mu]+ 1),(2*j + \[Lambda]- \[Mu]+ 1)*(2*j + \[Lambda]- \[Mu]+ 2)] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle M_{-1}P_{-1} = 0}
M_{-1}P_{-1} = 0 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | M[- 1]*P[- 1] = 0 |
Subscript[M, - 1]*Subscript[P, - 1] == 0 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lambda = \alpha}
\lambda = \alpha |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | lambda = alpha |
\[Lambda] == \[Alpha] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mu = \beta-\epsilon}
\mu = \beta-\epsilon |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | mu = beta - epsilon |
\[Mu] == \[Beta]- \[Epsilon] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lambda = \beta}
\lambda = \beta |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | lambda = beta |
\[Lambda] == \[Beta] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mu = \alpha-\epsilon}
\mu = \alpha-\epsilon |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | mu = alpha - epsilon |
\[Mu] == \[Alpha]- \[Epsilon] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11#Ex5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lambda = \gamma+\delta-1}
\lambda = \gamma+\delta-1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | lambda = gamma + delta - 1 |
\[Lambda] == \[Gamma]+ \[Delta]- 1 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11#Ex6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mu = 0}
\mu = 0 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | mu = 0 |
\[Mu] == 0 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11#Ex7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lambda = \gamma}
\lambda = \gamma |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | lambda = gamma |
\[Lambda] == \[Gamma] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11#Ex8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mu = \delta-1}
\mu = \delta-1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | mu = delta - 1 |
\[Mu] == \[Delta]- 1 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11#Ex9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lambda = \delta}
\lambda = \delta |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | lambda = delta |
\[Lambda] == \[Delta] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11#Ex10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mu = \gamma-1}
\mu = \gamma-1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | mu = gamma - 1 |
\[Mu] == \[Gamma]- 1 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11#Ex11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lambda = 1}
\lambda = 1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | lambda = 1 |
\[Lambda] == 1 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 31.11#Ex12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mu = \gamma+\delta-2}
\mu = \gamma+\delta-2 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | mu = gamma + delta - 2 |
\[Mu] == \[Gamma]+ \[Delta]- 2 |
Skipped - no semantic math | Skipped - no semantic math | - | - |