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 L 0 ⁒ c 0 + M 0 ⁒ c 1 = 0 subscript 𝐿 0 subscript 𝑐 0 subscript 𝑀 0 subscript 𝑐 1 0 {\displaystyle{\displaystyle L_{0}c_{0}+M_{0}c_{1}=0}}
L_{0}c_{0}+M_{0}c_{1} = 0		 

L[0]*c[0]+ M[0]*c[1] = 0
 
Subscript[L, 0]*Subscript[c, 0]+ Subscript[M, 0]*Subscript[c, 1] == 0
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31.11.E5 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 {\displaystyle{\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		 

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
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31.11.E6 K j = - ( j + Ξ± - ΞΌ - 1 ) ⁒ ( j + Ξ² - ΞΌ - 1 ) ⁒ ( j + Ξ³ - ΞΌ - 1 ) ⁒ ( j + Ξ» - 1 ) ( 2 ⁒ j + Ξ» - ΞΌ - 1 ) ⁒ ( 2 ⁒ j + Ξ» - ΞΌ - 2 ) subscript 𝐾 𝑗 𝑗 𝛼 πœ‡ 1 𝑗 𝛽 πœ‡ 1 𝑗 𝛾 πœ‡ 1 𝑗 πœ† 1 2 𝑗 πœ† πœ‡ 1 2 𝑗 πœ† πœ‡ 2 {\displaystyle{\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)}		 

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)]
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31.11.E7 L j = a ⁒ ( Ξ» + j ) ⁒ ( ΞΌ - j ) - q + ( j + Ξ± - ΞΌ ) ⁒ ( j + Ξ² - ΞΌ ) ⁒ ( j + Ξ³ - ΞΌ ) ⁒ ( j + Ξ» ) ( 2 ⁒ j + Ξ» - ΞΌ ) ⁒ ( 2 ⁒ j + Ξ» - ΞΌ + 1 ) + ( j - Ξ± + Ξ» ) ⁒ ( j - Ξ² + Ξ» ) ⁒ ( j - Ξ³ + Ξ» ) ⁒ ( j - ΞΌ ) ( 2 ⁒ j + Ξ» - ΞΌ ) ⁒ ( 2 ⁒ j + Ξ» - ΞΌ - 1 ) subscript 𝐿 𝑗 π‘Ž πœ† 𝑗 πœ‡ 𝑗 π‘ž 𝑗 𝛼 πœ‡ 𝑗 𝛽 πœ‡ 𝑗 𝛾 πœ‡ 𝑗 πœ† 2 𝑗 πœ† πœ‡ 2 𝑗 πœ† πœ‡ 1 𝑗 𝛼 πœ† 𝑗 𝛽 πœ† 𝑗 𝛾 πœ† 𝑗 πœ‡ 2 𝑗 πœ† πœ‡ 2 𝑗 πœ† πœ‡ 1 {\displaystyle{\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)}		 

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)]
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31.11.E8 M j = - ( j - Ξ± + Ξ» + 1 ) ⁒ ( j - Ξ² + Ξ» + 1 ) ⁒ ( j - Ξ³ + Ξ» + 1 ) ⁒ ( j - ΞΌ + 1 ) ( 2 ⁒ j + Ξ» - ΞΌ + 1 ) ⁒ ( 2 ⁒ j + Ξ» - ΞΌ + 2 ) subscript 𝑀 𝑗 𝑗 𝛼 πœ† 1 𝑗 𝛽 πœ† 1 𝑗 𝛾 πœ† 1 𝑗 πœ‡ 1 2 𝑗 πœ† πœ‡ 1 2 𝑗 πœ† πœ‡ 2 {\displaystyle{\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)}		 

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)]
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31.11.E9 M - 1 ⁒ P - 1 = 0 subscript 𝑀 1 subscript 𝑃 1 0 {\displaystyle{\displaystyle M_{-1}P_{-1}=0}}
M_{-1}P_{-1} = 0		 

M[- 1]*P[- 1] = 0
 
Subscript[M, - 1]*Subscript[P, - 1] == 0
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31.11#Ex1 Ξ» = Ξ± πœ† 𝛼 {\displaystyle{\displaystyle\lambda=\alpha}}
\lambda = \alpha		 

lambda = alpha
 
\[Lambda] == \[Alpha]
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31.11#Ex2 ΞΌ = Ξ² - Ο΅ πœ‡ 𝛽 italic-Ο΅ {\displaystyle{\displaystyle\mu=\beta-\epsilon}}
\mu = \beta-\epsilon		 

mu = beta - epsilon
 
\[Mu] == \[Beta]- \[Epsilon]
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31.11#Ex3 Ξ» = Ξ² πœ† 𝛽 {\displaystyle{\displaystyle\lambda=\beta}}
\lambda = \beta		 

lambda = beta
 
\[Lambda] == \[Beta]
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31.11#Ex4 ΞΌ = Ξ± - Ο΅ πœ‡ 𝛼 italic-Ο΅ {\displaystyle{\displaystyle\mu=\alpha-\epsilon}}
\mu = \alpha-\epsilon		 

mu = alpha - epsilon
 
\[Mu] == \[Alpha]- \[Epsilon]
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31.11#Ex5 Ξ» = Ξ³ + Ξ΄ - 1 πœ† 𝛾 𝛿 1 {\displaystyle{\displaystyle\lambda=\gamma+\delta-1}}
\lambda = \gamma+\delta-1		 

lambda = gamma + delta - 1
 
\[Lambda] == \[Gamma]+ \[Delta]- 1
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31.11#Ex6 ΞΌ = 0 πœ‡ 0 {\displaystyle{\displaystyle\mu=0}}
\mu = 0		 

mu = 0
 
\[Mu] == 0
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31.11#Ex7 Ξ» = Ξ³ πœ† 𝛾 {\displaystyle{\displaystyle\lambda=\gamma}}
\lambda = \gamma		 

lambda = gamma
 
\[Lambda] == \[Gamma]
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31.11#Ex8 ΞΌ = Ξ΄ - 1 πœ‡ 𝛿 1 {\displaystyle{\displaystyle\mu=\delta-1}}
\mu = \delta-1		 

mu = delta - 1
 
\[Mu] == \[Delta]- 1
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31.11#Ex9 Ξ» = Ξ΄ πœ† 𝛿 {\displaystyle{\displaystyle\lambda=\delta}}
\lambda = \delta		 

lambda = delta
 
\[Lambda] == \[Delta]
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31.11#Ex10 ΞΌ = Ξ³ - 1 πœ‡ 𝛾 1 {\displaystyle{\displaystyle\mu=\gamma-1}}
\mu = \gamma-1		 

mu = gamma - 1
 
\[Mu] == \[Gamma]- 1
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31.11#Ex11 Ξ» = 1 πœ† 1 {\displaystyle{\displaystyle\lambda=1}}
\lambda = 1		 

lambda = 1
 
\[Lambda] == 1
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31.11#Ex12 ΞΌ = Ξ³ + Ξ΄ - 2 πœ‡ 𝛾 𝛿 2 {\displaystyle{\displaystyle\mu=\gamma+\delta-2}}
\mu = \gamma+\delta-2		 

mu = gamma + delta - 2
 
\[Mu] == \[Gamma]+ \[Delta]- 2
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