Confluent hypergeometric functions

Symbol: Hypergeometric0F1 —

\,{}_0F_1\!\left(a, z\right)

— Confluent hypergeometric limit function

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric0F1`	$\,{}_0F_1\!\left(a, z\right)$	Confluent hypergeometric limit function

Source code for this entry:

Entry(ID("316533"),
    SymbolDefinition(Hypergeometric0F1, Hypergeometric0F1(a, z), "Confluent hypergeometric limit function"))

f565f5

Symbol: Hypergeometric0F1Regularized —

\,{}_0{\textbf F}_1\!\left(a, z\right)

— Regularized confluent hypergeometric limit function

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function

Source code for this entry:

Entry(ID("f565f5"),
    SymbolDefinition(Hypergeometric0F1Regularized, Hypergeometric0F1Regularized(a, z), "Regularized confluent hypergeometric limit function"))

512bea

Symbol: Hypergeometric1F1 —

\,{}_1F_1\!\left(a, b, z\right)

— Kummer confluent hypergeometric function

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function

Source code for this entry:

Entry(ID("512bea"),
    SymbolDefinition(Hypergeometric1F1, Hypergeometric1F1(a, b, z), "Kummer confluent hypergeometric function"))

cee331

Symbol: Hypergeometric1F1Regularized —

\,{}_1{\textbf F}_1\!\left(a, b, z\right)

— Regularized Kummer confluent hypergeometric function

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric1F1Regularized`	$\,{}_1{\textbf F}_1\!\left(a, b, z\right)$	Regularized Kummer confluent hypergeometric function

Source code for this entry:

Entry(ID("cee331"),
    SymbolDefinition(Hypergeometric1F1Regularized, Hypergeometric1F1Regularized(a, b, z), "Regularized Kummer confluent hypergeometric function"))

d6add6

Symbol: HypergeometricU —

U\!\left(a, b, z\right)

— Tricomi confluent hypergeometric function

Definitions:

Fungrim symbol	Notation	Short description
`HypergeometricU`	$U\!\left(a, b, z\right)$	Tricomi confluent hypergeometric function

Source code for this entry:

Entry(ID("d6add6"),
    SymbolDefinition(HypergeometricU, HypergeometricU(a, b, z), "Tricomi confluent hypergeometric function"))

1b9cc5

Symbol: HypergeometricUStar —

U^{*}\!\left(a, b, z\right)

— Scaled Tricomi confluent hypergeometric function

Definitions:

Fungrim symbol	Notation	Short description
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function

Source code for this entry:

Entry(ID("1b9cc5"),
    SymbolDefinition(HypergeometricUStar, HypergeometricUStar(a, b, z), "Scaled Tricomi confluent hypergeometric function"))

b9cc75

Symbol: Hypergeometric2F0 —

\,{}_2F_0\!\left(a, b, z\right)

— Tricomi confluent hypergeometric function, alternative notation

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric2F0`	$\,{}_2F_0\!\left(a, b, z\right)$	Tricomi confluent hypergeometric function, alternative notation

Source code for this entry:

Entry(ID("b9cc75"),
    SymbolDefinition(Hypergeometric2F0, Hypergeometric2F0(a, b, z), "Tricomi confluent hypergeometric function, alternative notation"))

4c41ad

\,{}_0F_1\!\left(a, z\right) = \sum_{k=0}^{\infty} \frac{1}{\left(a\right)_{k}} \frac{{z}^{k}}{k !}

Assumptions:

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

TeX:

\,{}_0F_1\!\left(a, z\right) = \sum_{k=0}^{\infty} \frac{1}{\left(a\right)_{k}} \frac{{z}^{k}}{k !}

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric0F1`	$\,{}_0F_1\!\left(a, z\right)$	Confluent hypergeometric limit function
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("4c41ad"),
    Formula(Equal(Hypergeometric0F1(a, z), Sum(Mul(Div(1, RisingFactorial(a, k)), Div(Pow(z, k), Factorial(k))), Tuple(k, 0, Infinity)))),
    Variables(a, z),
    Assumptions(And(Element(a, SetMinus(CC, ZZLessEqual(0))), Element(z, CC))))

0a0aec

\,{}_0{\textbf F}_1\!\left(a, z\right) = \sum_{k=0}^{\infty} \frac{1}{\Gamma\!\left(a + k\right)} \frac{{z}^{k}}{k !}

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

TeX:

\,{}_0{\textbf F}_1\!\left(a, z\right) = \sum_{k=0}^{\infty} \frac{1}{\Gamma\!\left(a + k\right)} \frac{{z}^{k}}{k !}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0a0aec"),
    Formula(Equal(Hypergeometric0F1Regularized(a, z), Sum(Mul(Div(1, GammaFunction(Add(a, k))), Div(Pow(z, k), Factorial(k))), Tuple(k, 0, Infinity)))),
    Variables(a, z),
    Assumptions(And(Element(a, CC), Element(z, CC))))

a61f01

\,{}_1F_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

TeX:

\,{}_1F_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("a61f01"),
    Formula(Equal(Hypergeometric1F1(a, b, z), Sum(Mul(Div(RisingFactorial(a, k), RisingFactorial(b, k)), Div(Pow(z, k), Factorial(k))), Tuple(k, 0, Infinity)))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, SetMinus(CC, ZZLessEqual(0))), Element(z, CC))))

dec042

\,{}_1F_1\!\left(-n, b, z\right) = \sum_{k=0}^{n} \frac{\left(-n\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{not} \left(b \in \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, b \gt -n\right) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

TeX:

\,{}_1F_1\!\left(-n, b, z\right) = \sum_{k=0}^{n} \frac{\left(-n\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\,  \operatorname{not} \left(b \in \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, b \gt -n\right) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("dec042"),
    Formula(Equal(Hypergeometric1F1(Neg(n), b, z), Sum(Mul(Div(RisingFactorial(Neg(n), k), RisingFactorial(b, k)), Div(Pow(z, k), Factorial(k))), Tuple(k, 0, n)))),
    Variables(n, b, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(b, CC), Not(And(Element(b, ZZLessEqual(0)), Greater(b, Neg(n)))), Element(z, CC))))

70111e

\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\Gamma\!\left(b + k\right)} \frac{{z}^{k}}{k !}

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

TeX:

\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\Gamma\!\left(b + k\right)} \frac{{z}^{k}}{k !}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric1F1Regularized`	$\,{}_1{\textbf F}_1\!\left(a, b, z\right)$	Regularized Kummer confluent hypergeometric function
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("70111e"),
    Formula(Equal(Hypergeometric1F1Regularized(a, b, z), Sum(Mul(Div(RisingFactorial(a, k), GammaFunction(Add(b, k))), Div(Pow(z, k), Factorial(k))), Tuple(k, 0, Infinity)))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC))))

06f229

z y''(z) + \left(b - z\right) y'(z) - a y\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = C \,{}_1{\textbf F}_1\!\left(a, b, z\right) + D U\!\left(a, b, z\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, C \in \mathbb{C} \,\mathbin{\operatorname{and}}\, D \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(D = 0 \,\mathbin{\operatorname{or}}\, z \ne 0 \,\mathbin{\operatorname{or}}\, -a \in \mathbb{Z}_{\ge 0}\right)

TeX:

z y''(z) + \left(b - z\right) y'(z) - a y\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = C \,{}_1{\textbf F}_1\!\left(a, b, z\right) + D U\!\left(a, b, z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, C \in \mathbb{C} \,\mathbin{\operatorname{and}}\, D \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(D = 0 \,\mathbin{\operatorname{or}}\, z \ne 0 \,\mathbin{\operatorname{or}}\, -a \in \mathbb{Z}_{\ge 0}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`Hypergeometric1F1Regularized`	$\,{}_1{\textbf F}_1\!\left(a, b, z\right)$	Regularized Kummer confluent hypergeometric function
`HypergeometricU`	$U\!\left(a, b, z\right)$	Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("06f229"),
    Formula(Where(Equal(Sub(Add(Mul(z, Derivative(y(z), Tuple(z, z, 2))), Mul(Sub(b, z), Derivative(y(z), Tuple(z, z, 1)))), Mul(a, y(z))), 0), Equal(y(z), Add(Mul(C, Hypergeometric1F1Regularized(a, b, z)), Mul(D, HypergeometricU(a, b, z)))))),
    Variables(z, a, b, C, D),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Element(C, CC), Element(D, CC), Or(Equal(D, 0), Unequal(z, 0), Element(Neg(a), ZZGreaterEqual(0))))))

bb5d67

z y''(z) + a y'(z) - y\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = C \,{}_0{\textbf F}_1\!\left(a, z\right) + D {z}^{1 - a} \,{}_0{\textbf F}_1\!\left(2 - a, z\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, C \in \mathbb{C} \,\mathbin{\operatorname{and}}\, D \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(D = 0 \,\mathbin{\operatorname{or}}\, z \ne 0 \,\mathbin{\operatorname{or}}\, 1 - a \in \mathbb{Z}_{\ge 0}\right)

TeX:

z y''(z) + a y'(z) - y\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = C \,{}_0{\textbf F}_1\!\left(a, z\right) + D {z}^{1 - a} \,{}_0{\textbf F}_1\!\left(2 - a, z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, C \in \mathbb{C} \,\mathbin{\operatorname{and}}\, D \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(D = 0 \,\mathbin{\operatorname{or}}\, z \ne 0 \,\mathbin{\operatorname{or}}\, 1 - a \in \mathbb{Z}_{\ge 0}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("bb5d67"),
    Formula(Where(Equal(Sub(Add(Mul(z, Derivative(y(z), Tuple(z, z, 2))), Mul(a, Derivative(y(z), Tuple(z, z, 1)))), y(z)), 0), Equal(y(z), Add(Mul(C, Hypergeometric0F1Regularized(a, z)), Mul(Mul(D, Pow(z, Sub(1, a))), Hypergeometric0F1Regularized(Sub(2, a), z)))))),
    Variables(z, a, C, D),
    Assumptions(And(Element(a, CC), Element(z, CC), Element(C, CC), Element(D, CC), Or(Equal(D, 0), Unequal(z, 0), Element(Sub(1, a), ZZGreaterEqual(0))))))

be533c

\,{}_1F_1\!\left(a, b, z\right) = {e}^{z} \,{}_1F_1\!\left(b - a, b, -z\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

TeX:

\,{}_1F_1\!\left(a, b, z\right) = {e}^{z} \,{}_1F_1\!\left(b - a, b, -z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("be533c"),
    Formula(Equal(Hypergeometric1F1(a, b, z), Mul(Exp(z), Hypergeometric1F1(Sub(b, a), b, Neg(z))))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, SetMinus(CC, ZZLessEqual(0))), Element(z, CC))))

a047eb

\,{}_1{\textbf F}_1\!\left(a, b, z\right) = {e}^{z} \,{}_1{\textbf F}_1\!\left(b - a, b, -z\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

TeX:

\,{}_1{\textbf F}_1\!\left(a, b, z\right) = {e}^{z} \,{}_1{\textbf F}_1\!\left(b - a, b, -z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric1F1Regularized`	$\,{}_1{\textbf F}_1\!\left(a, b, z\right)$	Regularized Kummer confluent hypergeometric function
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a047eb"),
    Formula(Equal(Hypergeometric1F1Regularized(a, b, z), Mul(Exp(z), Hypergeometric1F1Regularized(Sub(b, a), b, Neg(z))))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC))))

9d3147

U\!\left(a, b, z\right) = {z}^{1 - b} U\!\left(1 + a - b, 2 - b, z\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

TeX:

U\!\left(a, b, z\right) = {z}^{1 - b} U\!\left(1 + a - b, 2 - b, z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`HypergeometricU`	$U\!\left(a, b, z\right)$	Tricomi confluent hypergeometric function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9d3147"),
    Formula(Equal(HypergeometricU(a, b, z), Mul(Pow(z, Sub(1, b)), HypergeometricU(Sub(Add(1, a), b), Sub(2, b), z)))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Unequal(z, 0))))

c8fcc7

U^{*}\!\left(a, b, z\right) = {z}^{a} U\!\left(a, b, z\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

TeX:

U^{*}\!\left(a, b, z\right) = {z}^{a} U\!\left(a, b, z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`Pow`	${a}^{b}$	Power
`HypergeometricU`	$U\!\left(a, b, z\right)$	Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("c8fcc7"),
    Formula(Equal(HypergeometricUStar(a, b, z), Mul(Pow(z, a), HypergeometricU(a, b, z)))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Unequal(z, 0))))

4cf1e9

U^{*}\!\left(a, b, z\right) = \,{}_2F_0\!\left(a, a - b + 1, -\frac{1}{z}\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

TeX:

U^{*}\!\left(a, b, z\right) = \,{}_2F_0\!\left(a, a - b + 1, -\frac{1}{z}\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`Hypergeometric2F0`	$\,{}_2F_0\!\left(a, b, z\right)$	Tricomi confluent hypergeometric function, alternative notation
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4cf1e9"),
    Formula(Equal(HypergeometricUStar(a, b, z), Hypergeometric2F0(a, Add(Sub(a, b), 1), Neg(Div(1, z))))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Unequal(z, 0))))

f7f84e

\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma\!\left(b - a\right)} U^{*}\!\left(a, b, z\right) + \frac{{z}^{a - b} {e}^{z}}{\Gamma\!\left(a\right)} U^{*}\!\left(b - a, b, -z\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

TeX:

\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma\!\left(b - a\right)} U^{*}\!\left(a, b, z\right) + \frac{{z}^{a - b} {e}^{z}}{\Gamma\!\left(a\right)} U^{*}\!\left(b - a, b, -z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric1F1Regularized`	$\,{}_1{\textbf F}_1\!\left(a, b, z\right)$	Regularized Kummer confluent hypergeometric function
`Pow`	${a}^{b}$	Power
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f7f84e"),
    Formula(Equal(Hypergeometric1F1Regularized(a, b, z), Add(Mul(Div(Pow(Neg(z), Neg(a)), GammaFunction(Sub(b, a))), HypergeometricUStar(a, b, z)), Mul(Div(Mul(Pow(z, Sub(a, b)), Exp(z)), GammaFunction(a)), HypergeometricUStar(Sub(b, a), b, Neg(z)))))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Unequal(z, 0))))

6cf802

U\!\left(a, b, z\right) = \frac{\Gamma\!\left(1 - b\right)}{\Gamma\!\left(a - b + 1\right)} \,{}_1F_1\!\left(a, b, z\right) + \frac{\Gamma\!\left(b - 1\right)}{\Gamma\!\left(a\right)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, b \notin \mathbb{Z}

TeX:

U\!\left(a, b, z\right) = \frac{\Gamma\!\left(1 - b\right)}{\Gamma\!\left(a - b + 1\right)} \,{}_1F_1\!\left(a, b, z\right) + \frac{\Gamma\!\left(b - 1\right)}{\Gamma\!\left(a\right)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, b \notin \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`HypergeometricU`	$U\!\left(a, b, z\right)$	Tricomi confluent hypergeometric function
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("6cf802"),
    Formula(Equal(HypergeometricU(a, b, z), Add(Mul(Div(GammaFunction(Sub(1, b)), GammaFunction(Add(Sub(a, b), 1))), Hypergeometric1F1(a, b, z)), Mul(Mul(Div(GammaFunction(Sub(b, 1)), GammaFunction(a)), Pow(z, Sub(1, b))), Hypergeometric1F1(Add(Sub(a, b), 1), Sub(2, b), z))))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Unequal(z, 0), NotElement(b, ZZ))))

18ef23

U\!\left(a, n, z\right) = \lim_{b \to n} \frac{\Gamma\!\left(1 - b\right)}{\Gamma\!\left(a - b + 1\right)} \,{}_1F_1\!\left(a, b, z\right) + \frac{\Gamma\!\left(b - 1\right)}{\Gamma\!\left(a\right)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

TeX:

U\!\left(a, n, z\right) = \lim_{b \to n} \frac{\Gamma\!\left(1 - b\right)}{\Gamma\!\left(a - b + 1\right)} \,{}_1F_1\!\left(a, b, z\right) + \frac{\Gamma\!\left(b - 1\right)}{\Gamma\!\left(a\right)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`HypergeometricU`	$U\!\left(a, b, z\right)$	Tricomi confluent hypergeometric function
`ComplexLimit`	$\lim_{z \to a} f\!\left(z\right)$	Limiting value, complex variable
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("18ef23"),
    Formula(Equal(HypergeometricU(a, n, z), ComplexLimit(Add(Mul(Div(GammaFunction(Sub(1, b)), GammaFunction(Add(Sub(a, b), 1))), Hypergeometric1F1(a, b, z)), Mul(Mul(Div(GammaFunction(Sub(b, 1)), GammaFunction(a)), Pow(z, Sub(1, b))), Hypergeometric1F1(Add(Sub(a, b), 1), Sub(2, b), z))), b, n))),
    Variables(a, n, z),
    Assumptions(And(Element(a, CC), Element(n, ZZ), Element(z, CC), Unequal(z, 0))))

2df3e3

\,{}_0F_1\!\left(a, z\right) = {e}^{-2 \sqrt{z}} \,{}_1F_1\!\left(a - \frac{1}{2}, 2 a - 1, 4 \sqrt{z}\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, a \notin \{0, -1, \ldots\}

TeX:

\,{}_0F_1\!\left(a, z\right) = {e}^{-2 \sqrt{z}} \,{}_1F_1\!\left(a - \frac{1}{2}, 2 a - 1, 4 \sqrt{z}\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, a \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric0F1`	$\,{}_0F_1\!\left(a, z\right)$	Confluent hypergeometric limit function
`Exp`	${e}^{z}$	Exponential function
`Sqrt`	$\sqrt{z}$	Principal square root
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("2df3e3"),
    Formula(Equal(Hypergeometric0F1(a, z), Mul(Exp(Neg(Mul(2, Sqrt(z)))), Hypergeometric1F1(Sub(a, Div(1, 2)), Sub(Mul(2, a), 1), Mul(4, Sqrt(z)))))),
    Variables(a, z),
    Assumptions(And(Element(a, CC), Element(z, CC), NotElement(a, ZZLessEqual(0)))))

325a0e

\,{}_0{\textbf F}_1\!\left(a, z\right) = {\left(-z\right)}^{\left( 1 - a \right) / 2} J_{a - 1}\!\left(2 \sqrt{-z}\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

TeX:

\,{}_0{\textbf F}_1\!\left(a, z\right) = {\left(-z\right)}^{\left( 1 - a \right) / 2} J_{a - 1}\!\left(2 \sqrt{-z}\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`Pow`	${a}^{b}$	Power
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("325a0e"),
    Formula(Equal(Hypergeometric0F1Regularized(a, z), Mul(Pow(Neg(z), Div(Sub(1, a), 2)), BesselJ(Sub(a, 1), Mul(2, Sqrt(Neg(z))))))),
    Variables(a, z),
    Assumptions(And(Element(a, CC), Element(z, CC), Unequal(z, 0))))

00dfd1

\,{}_0{\textbf F}_1\!\left(a, z\right) = {z}^{\left( 1 - a \right) / 2} I_{a - 1}\!\left(2 \sqrt{z}\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

TeX:

\,{}_0{\textbf F}_1\!\left(a, z\right) = {z}^{\left( 1 - a \right) / 2} I_{a - 1}\!\left(2 \sqrt{z}\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`Pow`	${a}^{b}$	Power
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("00dfd1"),
    Formula(Equal(Hypergeometric0F1Regularized(a, z), Mul(Pow(z, Div(Sub(1, a), 2)), BesselI(Sub(a, 1), Mul(2, Sqrt(z)))))),
    Variables(a, z),
    Assumptions(And(Element(a, CC), Element(z, CC), Unequal(z, 0))))

d1b3b5

U^{*}\!\left(a, b, z\right) = \sum_{k=0}^{n - 1} \frac{\left(a\right)_{k} \left(a - b + 1\right)_{k}}{k ! {\left(-z\right)}^{k}} + R_{n}\!\left(a,b,z\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0}

TeX:

U^{*}\!\left(a, b, z\right) = \sum_{k=0}^{n - 1} \frac{\left(a\right)_{k} \left(a - b + 1\right)_{k}}{k ! {\left(-z\right)}^{k}} + R_{n}\!\left(a,b,z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`HypergeometricUStarRemainder`	$R_{n}\!\left(a,b,z\right)$	Error term in asymptotic expansion of Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("d1b3b5"),
    Formula(Equal(HypergeometricUStar(a, b, z), Add(Sum(Div(Mul(RisingFactorial(a, k), RisingFactorial(Add(Sub(a, b), 1), k)), Mul(Factorial(k), Pow(Neg(z), k))), Tuple(k, 0, Sub(n, 1))), HypergeometricUStarRemainder(n, a, b, z)))),
    Variables(a, b, z, n),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Unequal(z, 0), Element(n, ZZGreaterEqual(0)))))

99f69c

Symbol: HypergeometricUStarRemainder —

R_{n}\!\left(a,b,z\right)

— Error term in asymptotic expansion of Tricomi confluent hypergeometric function

Definitions:

Fungrim symbol	Notation	Short description
`HypergeometricUStarRemainder`	$R_{n}\!\left(a,b,z\right)$	Error term in asymptotic expansion of Tricomi confluent hypergeometric function

Source code for this entry:

Entry(ID("99f69c"),
    SymbolDefinition(HypergeometricUStarRemainder, HypergeometricUStarRemainder(n, a, b, z), "Error term in asymptotic expansion of Tricomi confluent hypergeometric function"))

876844

\lim_{z \to \infty} \left|R_{n}\!\left(a,b,{e}^{i \theta} z\right)\right| = 0

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \theta \in \mathbb{R} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 1}

TeX:

\lim_{z \to \infty} \left|R_{n}\!\left(a,b,{e}^{i \theta} z\right)\right| = 0

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \theta \in \mathbb{R} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexLimit`	$\lim_{z \to a} f\!\left(z\right)$	Limiting value, complex variable
`Abs`	$\left\|z\right\|$	Absolute value
`HypergeometricUStarRemainder`	$R_{n}\!\left(a,b,z\right)$	Error term in asymptotic expansion of Tricomi confluent hypergeometric function
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`RR`	$\mathbb{R}$	Real numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("876844"),
    Formula(Equal(ComplexLimit(Abs(HypergeometricUStarRemainder(n, a, b, Mul(Exp(Mul(ConstI, theta)), z))), z, Infinity), 0)),
    Variables(a, b, theta, n),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(theta, RR), Element(n, ZZGreaterEqual(1)))))

279e4f

\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2}{1 - \sigma} \exp\!\left(\frac{2 \rho}{\left(1 - \sigma\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\,\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\sigma \left(1 + \frac{\sigma}{4}\right)}{{\left(1 - \sigma\right)}^{2}}

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) \gt \left|b - 2 a\right|

References:

DLMF section 13.7, https://dlmf.nist.gov/13.7

TeX:

\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2}{1 - \sigma} \exp\!\left(\frac{2 \rho}{\left(1 - \sigma\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\,\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\sigma \left(1 + \frac{\sigma}{4}\right)}{{\left(1 - \sigma\right)}^{2}}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) \gt \left|b - 2 a\right|

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`HypergeometricUStarRemainder`	$R_{n}\!\left(a,b,z\right)$	Error term in asymptotic expansion of Tricomi confluent hypergeometric function
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("279e4f"),
    Formula(Where(LessEqual(Abs(HypergeometricUStarRemainder(n, a, b, z)), Mul(Mul(Abs(Div(Mul(RisingFactorial(a, n), RisingFactorial(Add(Sub(a, b), 1), n)), Mul(Factorial(n), Pow(z, n)))), Div(2, Sub(1, sigma))), Exp(Div(Mul(2, rho), Mul(Sub(1, sigma), Abs(z)))))), Equal(sigma, Div(Abs(Sub(b, Mul(2, a))), Abs(z))), Equal(rho, Add(Abs(Add(Sub(Pow(a, 2), Mul(a, b)), Div(b, 2))), Div(Mul(sigma, Add(1, Div(sigma, 4))), Pow(Sub(1, sigma), 2)))))),
    Variables(a, b, z, n),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Unequal(z, 0), Element(n, ZZGreaterEqual(0)), Greater(Re(z), Abs(Sub(b, Mul(2, a)))))),
    References("DLMF section 13.7, https://dlmf.nist.gov/13.7"))

461a54

\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2 \sqrt{1 + \frac{1}{2} \pi n}}{1 - \sigma} \exp\!\left(\frac{\pi \rho}{\left(1 - \sigma\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\,\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\sigma \left(1 + \frac{\sigma}{4}\right)}{{\left(1 - \sigma\right)}^{2}}

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \left(\left|\operatorname{Im}\!\left(z\right)\right| \gt \left|b - 2 a\right| \,\mathbin{\operatorname{or}}\, \operatorname{Re}\!\left(z\right) \gt \left|b - 2 a\right|\right)

References:

DLMF section 13.7, https://dlmf.nist.gov/13.7

TeX:

\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2 \sqrt{1 + \frac{1}{2} \pi n}}{1 - \sigma} \exp\!\left(\frac{\pi \rho}{\left(1 - \sigma\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\,\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\sigma \left(1 + \frac{\sigma}{4}\right)}{{\left(1 - \sigma\right)}^{2}}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \left(\left|\operatorname{Im}\!\left(z\right)\right| \gt \left|b - 2 a\right| \,\mathbin{\operatorname{or}}\, \operatorname{Re}\!\left(z\right) \gt \left|b - 2 a\right|\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`HypergeometricUStarRemainder`	$R_{n}\!\left(a,b,z\right)$	Error term in asymptotic expansion of Tricomi confluent hypergeometric function
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("461a54"),
    Formula(Where(LessEqual(Abs(HypergeometricUStarRemainder(n, a, b, z)), Mul(Mul(Abs(Div(Mul(RisingFactorial(a, n), RisingFactorial(Add(Sub(a, b), 1), n)), Mul(Factorial(n), Pow(z, n)))), Div(Mul(2, Sqrt(Add(1, Mul(Mul(Div(1, 2), ConstPi), n)))), Sub(1, sigma))), Exp(Div(Mul(ConstPi, rho), Mul(Sub(1, sigma), Abs(z)))))), Equal(sigma, Div(Abs(Sub(b, Mul(2, a))), Abs(z))), Equal(rho, Add(Abs(Add(Sub(Pow(a, 2), Mul(a, b)), Div(b, 2))), Div(Mul(sigma, Add(1, Div(sigma, 4))), Pow(Sub(1, sigma), 2)))))),
    Variables(a, b, z, n),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Unequal(z, 0), Element(n, ZZGreaterEqual(0)), Or(Greater(Abs(Im(z)), Abs(Sub(b, Mul(2, a)))), Greater(Re(z), Abs(Sub(b, Mul(2, a))))))),
    References("DLMF section 13.7, https://dlmf.nist.gov/13.7"))

7b91b4

\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2 C\!\left(n\right)}{1 - \tau} \exp\!\left(\frac{2 C\!\left(1\right) \rho}{\left(1 - \tau\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\,\nu = 1 + 2 {\sigma}^{2},\,\tau = \nu \sigma,\,\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\tau \left(1 + \frac{\tau}{4}\right)}{{\left(1 - \tau\right)}^{2}},\,C\!\left(m\right) = \left(\sqrt{1 + \frac{\pi m}{2}} + \sigma {\nu}^{2} m\right) {\nu}^{m}

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \left|z\right| \gt 2 \left|b - 2 a\right|

References:

DLMF section 13.7, https://dlmf.nist.gov/13.7

TeX:

\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2 C\!\left(n\right)}{1 - \tau} \exp\!\left(\frac{2 C\!\left(1\right) \rho}{\left(1 - \tau\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\,\nu = 1 + 2 {\sigma}^{2},\,\tau = \nu \sigma,\,\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\tau \left(1 + \frac{\tau}{4}\right)}{{\left(1 - \tau\right)}^{2}},\,C\!\left(m\right) = \left(\sqrt{1 + \frac{\pi m}{2}} + \sigma {\nu}^{2} m\right) {\nu}^{m}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \left|z\right| \gt 2 \left|b - 2 a\right|

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`HypergeometricUStarRemainder`	$R_{n}\!\left(a,b,z\right)$	Error term in asymptotic expansion of Tricomi confluent hypergeometric function
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstPi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("7b91b4"),
    Formula(Where(LessEqual(Abs(HypergeometricUStarRemainder(n, a, b, z)), Mul(Abs(Div(Mul(RisingFactorial(a, n), RisingFactorial(Add(Sub(a, b), 1), n)), Mul(Factorial(n), Pow(z, n)))), Mul(Div(Mul(2, C(n)), Sub(1, tau)), Exp(Div(Mul(Mul(2, C(1)), rho), Mul(Sub(1, tau), Abs(z))))))), Equal(sigma, Div(Abs(Sub(b, Mul(2, a))), Abs(z))), Equal(nu, Add(1, Mul(2, Pow(sigma, 2)))), Equal(tau, Mul(nu, sigma)), Equal(rho, Add(Abs(Add(Sub(Pow(a, 2), Mul(a, b)), Div(b, 2))), Div(Mul(tau, Add(1, Div(tau, 4))), Pow(Sub(1, tau), 2)))), Equal(C(m), Mul(Add(Sqrt(Add(1, Div(Mul(ConstPi, m), 2))), Mul(Mul(sigma, Pow(nu, 2)), m)), Pow(nu, m))))),
    Variables(a, b, z, n),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Unequal(z, 0), Element(n, ZZGreaterEqual(0)), Greater(Abs(z), Mul(2, Abs(Sub(b, Mul(2, a))))))),
    References("DLMF section 13.7, https://dlmf.nist.gov/13.7"))

Hypergeometric series

Differential equations

Kummer's transformation

Connection formulas

Asymptotic expansions