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 Hypergeometric1F1

 http://functions.wolfram.com/07.20.03.0113.01

 Input Form

 Hypergeometric1F1[a, 2 a - n, z] == Gamma[a + 1/2] E^(z/2) (z/4)^(1/2 - a) BesselI[a - 1/2, z/2] Sum[((k - j)! (-1)^(n - k - j) Pochhammer[-n, n - k] Pochhammer[2 a - 2 n - 1, n - k] 2^(-2 n + 2 k - 4 j) z^(n - k + 2 j))/(j! (k - 2 j)! (n - k)! Pochhammer[-a + 1/2, j] Pochhammer[a - k + 1/2, j] Pochhammer[2 a - n, n - k] Pochhammer[a - n - 1/2, n - k]), {k, 0, n}, {j, 0, Floor[k/2]}] + Gamma[a - 1/2] E^(z/2) (z/4)^(3/2 - a) BesselI[a + 1/2, z/2] Sum[((k - j - 1)! (-1)^(n - k - j) Pochhammer[-n, n - k] Pochhammer[2 a - 2 n - 1, n - k] 2^(-2 n + 2 k - 4 j) z^(n - k + 2 j))/(j! (k - 2 j - 1)! (n - k)! Pochhammer[3/2 - a, j] Pochhammer[a - k + 1/2, j] Pochhammer[2 a - n, n - k] Pochhammer[a - n - 1/2, n - k]), {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]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[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 / 2 ( z 4 ) 1 2 - a ( k = 0 n j = 0 k 2 ( - 1 ) n - j - k ( k - j ) ! ( - n ) n - k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], RowBox[List["n", "-", "k"]]], Pochhammer] ( 2 a - 2 n - 1 ) n - k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", RowBox[List["2", " ", "n"]], "-", "1"]], ")"]], RowBox[List["n", "-", "k"]]], Pochhammer] 2 - 4 j + 2 k - 2 n z 2 j - k + n j ! ( k - 2 j ) ! ( n - k ) ! ( 1 2 - a ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "a"]], ")"]], "j"], Pochhammer] ( a - k + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "k", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] ( 2 a - n ) n - k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", "n"]], ")"]], RowBox[List["n", "-", "k"]]], Pochhammer] ( a - n - 1 2 ) n - k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "n", "-", FractionBox["1", "2"]]], ")"]], RowBox[List["n", "-", "k"]]], 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 ) n - j - k ( k - j - 1 ) ! ( - n ) n - k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], RowBox[List["n", "-", "k"]]], Pochhammer] ( 2 a - 2 n - 1 ) n - k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", RowBox[List["2", " ", "n"]], "-", "1"]], ")"]], RowBox[List["n", "-", "k"]]], Pochhammer] 2 - 4 j + 2 k - 2 n z 2 j - k + n j ! ( k - 2 j - 1 ) ! ( n - k ) ! ( 3 2 - a ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["3", "2"], "-", "a"]], ")"]], "j"], Pochhammer] ( a - k + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "k", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] ( 2 a - n ) n - k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", "n"]], ")"]], RowBox[List["n", "-", "k"]]], Pochhammer] ( a - n - 1 2 ) n - k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "n", "-", FractionBox["1", "2"]]], ")"]], RowBox[List["n", "-", "k"]]], Pochhammer] ) I a + 1 2 ( z 2 ) /; n Condition Hypergeometric1F1 a 2 a -1 n z Gamma a 1 2 z 2 -1 z 4 -1 1 2 -1 a j 0 k 2 -1 k 0 n -1 n -1 j -1 k k -1 j Pochhammer -1 n n -1 k Pochhammer 2 a -1 2 n -1 n -1 k 2 -4 j 2 k -1 2 n z 2 j -1 k n j k -1 2 j n -1 k Pochhammer 1 2 -1 a j Pochhammer a -1 k 1 2 j Pochhammer 2 a -1 n n -1 k Pochhammer a -1 n -1 1 2 n -1 k -1 BesselI a -1 1 2 z 2 -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 n -1 j -1 k k -1 j -1 Pochhammer -1 n n -1 k Pochhammer 2 a -1 2 n -1 n -1 k 2 -4 j 2 k -1 2 n z 2 j -1 k n j k -1 2 j -1 n -1 k Pochhammer 3 2 -1 a j Pochhammer a -1 k 1 2 j Pochhammer 2 a -1 n n -1 k Pochhammer a -1 n -1 1 2 n -1 k -1 BesselI a 1 2 z 2 -1 n [/itex]

 Rule Form

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

 2007-05-02