Generalized hypergeometric function: Series representations (formula 07.31.06.0006)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]

Series representations

Generalized power series

Expansions at z==1 for p==q+1

The logarithmic cases

http://functions.wolfram.com/07.31.06.0006.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == (Product[Gamma[Subscript[b, k]], {k, 1, q}]/Product[Gamma[Subscript[a, k]], {k, 1, q + 1}]) (Sum[Subscript[k, j] (1 - j)^j, {j, 0, Subscript[\[Psi], q] - 1}] + (1 - z)^Subscript[\[Psi], q] Sum[(Subscript[p, j] + Subscript[q, j] Log[1 - z]) (1 - z)^j, {j, 0, Infinity}]) /; Abs[1 - z] < 1 && Subscript[\[Psi], q] == Sum[Subscript[b, j], {j, 1, q}] - Sum[Subscript[a, j], {j, 1, q + 1}] && q > 1 && (Subscript[k, j] == (((-1)^j Gamma[Subscript[a, 1] + j] Gamma[Subscript[a, 2] + j])/j!) Sum[((Subscript[\[Psi], q] + k - j - 1)!/ (Gamma[Subscript[a, 1] + Subscript[\[Psi], q] + k] Gamma[Subscript[a, 2] + Subscript[\[Psi], q] + k])) HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, k], {k, 0, Infinity}] /; Re[Subscript[a, 3]] > -j && \[Ellipsis] && Re[Subscript[a, p + 1]] > -j) && (Subscript[p, j] == (((-1)^Subscript[\[Psi], q] Pochhammer[Subscript[a, 1] + Subscript[\[Psi], q], j] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], q], j])/(j! (Subscript[\[Psi], q] + j)!)) ((-1)^j j! Sum[((k - j - 1)!/(Pochhammer[Subscript[a, 1] + Subscript[\[Psi], q], k] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], q], k])) HypergeometricPFQExpansionCoefficient[ {Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, k], {k, j + 1, Infinity}] + Sum[(Pochhammer[-j, k]/(Pochhammer[Subscript[a, 1] + Subscript[\[Psi], q], k] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], q], k])) HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, k] (PolyGamma[j - k + 1] + PolyGamma[j + Subscript[\[Psi], q] + 1] - PolyGamma[Subscript[\[Psi], q] + Subscript[a, 1] + j] - PolyGamma[Subscript[\[Psi], q] + Subscript[a, 2] + j]), {k, 0, j}]) /; Re[Subscript[a, 3]] > -j - Subscript[\[Psi], q] && \[Ellipsis] && Re[Subscript[a, p + 1]] > -j - Subscript[\[Psi], q]) && Subscript[q, j] == (((-1)^(Subscript[\[Psi], q] - 1) Pochhammer[Subscript[a, 1] + Subscript[\[Psi], q], j] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], q], j])/ (j! (Subscript[\[Psi], q] + j)!)) Sum[(Pochhammer[-j, k]/(Pochhammer[Subscript[a, 1] + Subscript[\[Psi], q], k] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], q], k])) HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, k], {k, 0, j}] && Element[Subscript[\[Psi], q], Integers] && Subscript[\[Psi], q] >= 0

Standard Form

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MathML Form

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</mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> ℰ </mi> <mi> k </mi> <mrow> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mi> j </mi> </mrow> </mrow> <mo> ∧ </mo> <mo> … </mo> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mrow> <mi> p </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mi> j </mi> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> p </mi> <mi> j </mi> </msub> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "j"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "j"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> ℰ </mi> <mi> k </mi> <mrow> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> </mfrac> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> j </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "j"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <msubsup> <mi> ℰ </mi> <mi> k </mi> <mrow> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> - </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> </mrow> <mo> ∧ </mo> <mo> … </mo> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mrow> <mi> p </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> - </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> q </mi> <mi> j </mi> </msub> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> ψ </mi> <mi> q </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "j"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "j"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> j </mi> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "j"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <msubsup> <mi> ℰ </mi> <mi> k </mi> <mrow> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> ψ </mi> <mi> q </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> ψ </mi> <mi> q </mi> </msub> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> q </ci> </apply> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> j </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <ci> j </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> q </ci> <ci> j </ci> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> j </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> j </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> j </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ℰ </ci> <ci> k </ci> </apply> <ci> q </ci> </apply> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> q </ci> </apply> </list> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <ci> … </ci> <apply> <gt /> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> j </ci> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <factorial /> <ci> j </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ℰ </ci> <ci> k </ci> </apply> <ci> q </ci> </apply> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> q </ci> </apply> </list> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> j </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <ci> k </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ℰ </ci> <ci> k </ci> </apply> <ci> q </ci> </apply> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> q </ci> </apply> </list> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> j </ci> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> j </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> j </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> </apply> </apply> <ci> … </ci> <apply> <gt /> <apply> <real /> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> q </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> j </ci> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> j </ci> </uplimit> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <ci> k </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ℰ </ci> <ci> k </ci> </apply> <ci> q </ci> </apply> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> q </ci> </apply> </list> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> </apply> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> ψ </ci> <ci> q </ci> </apply> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Date Added to functions.wolfram.com (modification date)

2001-10-29

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]