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 Hypergeometric1F1

 http://functions.wolfram.com/07.20.03.0114.01

 Input Form

 Hypergeometric1F1[a, 2 a + n, z] == Gamma[a + 1/2] (z/4)^(-a + 1/2) E^(z/2) BesselI[a - 1/2, z/2] Sum[((-1)^(k - j) 2^(2 k - 4 j) z^(2 j - k) (k - j)! Pochhammer[1/2 + a, k] Pochhammer[2 a - 1, k] Pochhammer[-n, k])/(j! k! (k - 2 j)! Pochhammer[2 a + n, k] Pochhammer[3/2 - a - k, j] Pochhammer[a - 1/2, j]), {k, 0, n}, {j, 0, Floor[k/2]}] - Gamma[-(1/2) + a] E^(z/2) (z/4)^(-a + 3/2) BesselI[a - 3/2, z/2] Sum[((-1)^(k - j) 2^(2 k - 4 j) z^(2 j - k) (k - j - 1)! Pochhammer[1/2 + a, k] Pochhammer[-1 + 2 a, k] Pochhammer[-n, k])/(j! k! (k - 2 j - 1)! Pochhammer[2 a + n, k] Pochhammer[3/2 - a - k, j] Pochhammer[1/2 + a, j]), {k, 0, n}, {j, 0, Floor[(k - 1)/2]}] /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 1 F 1 ( a ; 2 a + n ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["a", Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", "n"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] Γ ( a + 1 2 ) ( z 4 ) 1 2 - a z / 2 ( k = 0 n j = 0 k 2 ( - 1 ) k - j 2 2 k - 4 j z 2 j - k ( k - j ) ! ( a + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( 2 a - 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", "1"]], ")"]], "k"], Pochhammer] ( - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "k"], Pochhammer] j ! k ! ( k - 2 j ) ! ( 2 a + n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "n"]], ")"]], "k"], Pochhammer] ( 3 2 - a - k ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["3", "2"], "-", "a", "-", "k"]], ")"]], "j"], Pochhammer] ( a - 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] ) I a - 1 2 ( z 2 ) - Γ ( a - 1 2 ) z / 2 ( z 4 ) 3 2 - a ( k = 0 n j = 0 k - 1 2 ( - 1 ) k - j 2 2 k - 4 j z 2 j - k ( k - j - 1 ) ! ( a + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( 2 a - 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", "1"]], ")"]], "k"], Pochhammer] ( - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "k"], Pochhammer] j ! k ! ( k - 2 j - 1 ) ! ( 2 a + n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "n"]], ")"]], "k"], Pochhammer] ( 3 2 - a - k ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["3", "2"], "-", "a", "-", "k"]], ")"]], "j"], Pochhammer] ( a + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] ) I a - 3 2 ( z 2 ) /; n Condition Hypergeometric1F1 a 2 a n z Gamma a 1 2 z 4 -1 1 2 -1 a z 2 -1 j 0 k 2 -1 k 0 n -1 k -1 j 2 2 k -1 4 j z 2 j -1 k k -1 j Pochhammer a 1 2 k Pochhammer 2 a -1 k Pochhammer -1 n k j k k -1 2 j Pochhammer 2 a n k Pochhammer 3 2 -1 a -1 k j Pochhammer a -1 1 2 j -1 BesselI a -1 1 2 z 2 -1 -1 Gamma a -1 1 2 z 2 -1 z 4 -1 3 2 -1 a j 0 k -1 2 -1 k 0 n -1 k -1 j 2 2 k -1 4 j z 2 j -1 k k -1 j -1 Pochhammer a 1 2 k Pochhammer 2 a -1 k Pochhammer -1 n k j k k -1 2 j -1 Pochhammer 2 a n k Pochhammer 3 2 -1 a -1 k j Pochhammer a 1 2 j -1 BesselI a -1 3 2 z 2 -1 n [/itex]

 Rule Form

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

 2007-05-02