aarjaneiro/planetmath huggingface dataset:
generating function of Laguerre polynomials

generating function of Laguerre polynomials

We start from the definition of

Laguerre polynomials

via their

http://planetmath.org/node/11983

Rodrigues formula

L

n

(

z

)

:=

e

z

d

n

d

⁢

z

n

e

-

z

z

n

(

n

=

0

,

1

,

2

,

…

)

.

(1)

The consequence

f

(

n

)

⁢

(

z

)

=

n

!

2

⁢

π

⁢

i

⁢

∮

C

f

⁢

(

ζ

)

(

ζ

-

z

)

n

+

1

⁢

𝑑

ζ

(2)

of

http://planetmath.org/node/1150

Cauchy integral formula

allows to write (1) as the

complex integral

L

n

⁢

(

z

)

=

n

!

2

⁢

i

⁢

π

⁢

∮

C

e

z

⁢

e

-

ζ

(

ζ

-

z

)

n

+

1

⁢

𝑑

ζ

=

n

!

2

⁢

i

⁢

π

⁢

∮

C

e

z

-

ζ

⁢

d

⁢

ζ

(

1

-

z

ζ

)

n

⁢

(

ζ

-

z

)

,

where

C

is any contour around the point

z

and the direction is anticlockwise.  The

http://planetmath.org/node/11373

substitution

ζ

-

z

:=

z

⁢

t

1

-

t

,

ζ

=

z

1

-

t

,

t

=

1

-

z

ζ

d

⁢

ζ

=

z

⁢

d

⁢

t

(

1

-

t

)

2

here yields

L

n

⁢

(

z

)

=

n

!

2

⁢

i

⁢

π

⁢

∮

C

′

e

-

z

⁢

t

1

-

t

⁢

z

⁢

d

⁢

t

(

1

-

t

)

2

⁢

t

n

⋅

z

⁢

t

1

-

t

=

n

!

2

⁢

i

⁢

π

⁢

∮

C

′

e

-

z

⁢

t

1

-

t

⁢

d

⁢

t

(

1

-

t

)

⁢

t

n

+

1

where the contour

C

′

goes round the origin.  Accordingly, by (2) we can infer that

L

n

⁢

(

z

)

=

[

d

n

d

⁢

t

n

⁢

e

-

z

⁢

t

1

-

t

1

-

t

]

t

=

0

,

whence we have found the

generating function

e

-

z

⁢

t

1

-

t

1

-

t

=

∑

n

=

0

∞

L

n

⁢

(

z

)

n

!

⁢

t

n

of the Laguerre polynomials.

Title

generating function of Laguerre polynomials

Canonical name

GeneratingFunctionOfLaguerrePolynomials

Date of creation

2013-03-22 19:06:51

Last modified on

2013-03-22 19:06:51

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Derivation

Classification

msc 33B99

Classification

msc 30B10

Classification

msc 26C05

Classification

msc 26A09

Classification

msc 33E30

Related topic

ExampleOfFindingTheGeneratingFunction

Related topic

GeneratingFunctionOfHermitePolynomials

Related topic

VariantOfCauchyIntegralFormula

Generated on Fri Feb 9 20:36:09 2018 by

LaTeXML
generating function of Legendre polynomials

generating function of Legendre polynomials

For finding the

generating function

F

⁢

(

t

)

=

∑

n

=

0

∞

P

n

⁢

(

z

)

⁢

t

n

of the sequence of the

Legendre polynomials

P

0

⁢

(

z

)

=

1

P

1

⁢

(

z

)

=

z

P

2

⁢

(

z

)

=

1

2

⁢

(

3

⁢

z

2

-

1

)

P

3

⁢

(

x

)

=

1

2

⁢

(

5

⁢

z

3

-

3

⁢

z

)

P

4

⁢

(

z

)

=

1

8

⁢

(

35

⁢

z

4

-

30

⁢

z

2

+

3

)

P

5

⁢

(

z

)

=

1

8

⁢

(

63

⁢

z

5

-

70

⁢

z

3

+

15

⁢

z

)

⋯

⋯

we have to present

P

n

⁢

(

z

)

as the general coefficient of

Taylor series

in

t

,
i.e. as the

n

th derivative of some

F

⁢

(

t

)

in the origin, divided by the

factorial

n

!

.  The

Cauchy integral formula

offers the chance to implement that.

Starting from the

http://planetmath.org/node/11983

Rodrigues formula

of Legendre polynomials, we may write

P

n

⁢

(

z

)

=

1

2

n

⁢

n

!

⁢

d

n

d

⁢

z

n

⁢

(

z

2

-

1

)

n

=

1

2

n

⁢

n

!

⁢

n

!

2

⁢

i

⁢

π

⁢

∮

c

(

ζ

2

-

1

)

n

(

ζ

-

z

)

n

+

1

⁢

𝑑

ζ

=

1

2

⁢

i

⁢

π

⁢

∮

c

(

1

2

⁢

ζ

2

-

1

ζ

-

z

)

n

⁢

d

⁢

ζ

ζ

-

z

,

where the

contour

c

runs anticlockwise once around the point

z

.  The change of variable

ζ

2

-

1

2

⁢

(

ζ

-

z

)

=

1

t

,

d

⁢

ζ

=

z

⁢

t

-

1

-

1

-

z

⁢

t

+

t

2

t

2

⁢

1

-

z

⁢

t

+

t

2

⁢

d

⁢

t

gives

P

n

⁢

(

z

)

=

-

1

2

⁢

i

⁢

π

⁢

∮

c

′

d

⁢

t

t

n

⁢

t

⁢

1

-

z

⁢

t

+

t

2

where

t

must go round the origin clockwise, but in

P

n

⁢

(

z

)

=

1

n

!

⋅

n

!

2

⁢

i

⁢

π

⁢

∮

c

′

d

⁢

t

1

-

z

⁢

t

+

t

2

⋅

(

t

-

0

)

n

+

1

anticlockwise.  This is, by Cauchy integral formula again,

P

n

⁢

(

z

)

=

1

n

!

⁢

[

d

n

d

⁢

t

n

⁢

1

1

-

z

⁢

t

+

t

2

]

t

=

0

.

This means that

F

⁢

(

t

)

:=

1

1

-

z

⁢

t

+

t

2

is the searched generating function of the Legendre polynomials:

1

1

-

z

⁢

t

+

t

2

=

P

0

⁢

(

z

)

+

P

1

⁢

(

z

)

⁢

t

+

P

2

⁢

(

z

)

⁢

t

2

+

P

3

⁢

(

z

)

⁢

t

3

+

…

Cf. the generating function of the

Bessel functions

.

Title

generating function of Legendre polynomials

Canonical name

GeneratingFunctionOfLegendrePolynomials

Date of creation

2015-03-08 20:29:06

Last modified on

2015-03-08 20:29:06

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Result

Classification

msc 33B99

Classification

msc 30D10

Classification

msc 30B10

Related topic

GeneratingFunctionOfHermitePolynomials

Related topic

GeneratingFunctionOfLaguerrePolynomials

Related topic

VariantOfCauchyIntegralFormula

Generated on Fri Feb 9 20:36:12 2018 by

LaTeXML
natural log base

natural log base

The

natural log base

, or

e

, has value

2.718281828459045

⁢

…

e

was extensively studied by Euler in the 1720’s, but it was originally discovered by John Napier.

e

is defined by

lim

n

→

∞

⁡

(

1

+

1

n

)

n

It is more effectively calculated, however, by using the

Taylor series

for

f

⁢

(

x

)

=

e

x

at

x

=

1

to get the representation

e

=

1

0

!

+

1

1

!

+

1

2

!

+

1

3

!

+

1

4

!

+

⋯

Title

natural log base

Canonical name

NaturalLogBase

Date of creation

2013-03-22 11:55:56

Last modified on

2013-03-22 11:55:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 33B99

Synonym

Euler number

Synonym

Eulerian number

Synonym

Napier’s constant

Synonym

e

Related topic

ExampleOfTaylorPolynomialsForTheExponentialFunction

Related topic

EIsTranscendental

Related topic

EIsIrrationalProof

Related topic

ApplicationOfCauchyCriterionForConvergence

Generated on Fri Feb 9 20:36:14 2018 by

LaTeXML
proof of bounds for e

proof of bounds for e

Multiplying and dividing, we have

(

1

+

1

n

)

n

=

(

1

+

1

m

)

m

⁢

∏

k

=

m

+

1

n

(

1

+

1

k

)

k

(

1

+

1

k

-

1

)

k

-

1

As was shown in the parent entry, the quotients in the product can
be simplified to give

(

1

+

1

n

)

n

=

(

1

+

1

m

)

m

⁢

∏

k

=

m

+

1

n

(

1

-

1

k

2

)

k

⁢

(

1

+

1

k

-

1

)

By the

inequality for differences of powers

,

(

1

-

1

k

2

)

k

<

1

-

k

k

2

+

k

-

1

=

(

k

+

1

)

⁢

(

k

-

1

)

k

2

+

k

-

1

Hence, we have the following

upper bound

:

(

1

+

1

n

)

n

<

(

1

+

1

m

)

m

⁢

∏

k

=

m

+

1

n

k

2

+

k

k

2

+

k

-

1

By cross mutliplying, it is easy to see that

k

2

+

k

k

2

+

k

-

1

≤

k

2

k

2

-

1

and, hence,

(

1

+

1

n

)

n

<

(

1

+

1

m

)

m

⁢

∏

k

=

m

+

1

n

k

2

k

2

-

1

.

Factoring the

rational function

in the product, terms cancel and we
have

∏

k

=

m

+

1

n

k

2

(

k

+

1

)

⁢

(

k

-

1

)

=

n

⁢

(

m

+

1

)

(

n

+

1

)

⁢

m

=

n

n

+

1

⁢

(

1

+

1

m

)

Combining,

(

1

+

1

n

)

n

<

n

n

+

1

⁢

(

1

+

1

m

)

m

+

1

Title

proof of bounds for e

Canonical name

ProofOfBoundsForE

Date of creation

2013-03-22 15:48:51

Last modified on

2013-03-22 15:48:51

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 33B99

Generated on Fri Feb 9 20:36:16 2018 by

LaTeXML
values of (1+1/n)^n for 0<n<26

values of

(

1

+

1

/

n

)

n

for

0

<

n

<

26

The following table gives the

numerator

and denominator of

(

1

+

1

n

)

n

as well as the

decimal expansion

to 20 places.

n

Numerator of

(

1

+

1

n

)

n

Denominator of

(

1

+

1

n

)

n

Decimal value of

(

1

+

1

n

)

n

1

2

1

2.0000000000000000000

2

9

4

2.2500000000000000000

3

64

27

2.3703703703703703704

4

625

256

2.4414062500000000000

5

7776

3125

2.4883200000000000000

6

117649

46656

2.5216263717421124829

7

2097152

823543

2.5464996970407131139

8

43046721

16777216

2.5657845139503479004

9

1000000000

387420489

2.5811747917131971820

10

25937424601

10000000000

2.5937424601000000000

11

743008370688

285311670611

2.6041990118975308782

12

23298085122481

8916100448256

2.6130352902246781603

13

793714773254144

302875106592253

2.6206008878857322211

14

29192926025390625

11112006825558016

2.6271515563008693884

15

1152921504606846976

437893890380859375

2.6328787177279190470

16

48661191875666868481

18446744073709551616

2.6379284973665998588

17

2185911559738696531968

827240261886336764177

2.6424143751831096203

18

104127350297911241532841

39346408075296537575424

2.6464258210976854673

19

5242880000000000000000000

1978419655660313589123979

2.6500343266404449073

20

278218429446951548637196401

104857600000000000000000000

2.6532977051444201339

21

15519448971100888972574851072

5842587018385982521381124421

2.6562632139261049855

22

907846434775996175406740561329

341427877364219557396646723584

2.6589698585377882029

23

55572324035428505185378394701824

20880467999847912034355032910567

2.6614501186387814545

24

3552713678800500929355621337890625

1333735776850284124449081472843776

2.6637312580685940367

25

236773830007967588876795164938469376

88817841970012523233890533447265625

2.6658363314874199930

With a large enough value of

n

, this

formula

approximates the natural log base

e

. For example, with

n

set to ten million, we get 2.7182816925449662712, which is 0.000000135914078964161737245574 short of 2.7182818284590452354 (this calculation took almost four minutes with

Mathematica

4.2). It is less computationally intensive to use

∑

i

=

0

n

1

i

!

,

which with

n

set to 100 gives in less than a second a result to 20 places that is

indistinguishable

from

e

.

Title

values of

(

1

+

1

/

n

)

n

for

0

<

n

<

26

Canonical name

ValuesOf11nnFor0N26

Date of creation

2013-03-22 17:02:26

Last modified on

2013-03-22 17:02:26

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 33B99

Generated on Fri Feb 9 20:36:18 2018 by

LaTeXML
values of ∑@⁢ERROR(\slimits)⁢@⁢@⁢@_{i=0}^n⁢1/{i!} for 0<n<26

values of

∑

@

⁢

\slimits

⁢

@

⁢

@

⁢

@

i

=

0

n

⁢

1

i

!

for

0

<

n

<

26

The following table gives the

numerator

and denominator of

∑

i

=

0

n

1

i

!

as well as the

decimal expansion

to 20 places.

n

Numerator of

∑

i

=

0

n

1

i

!

Denominator of

∑

i

=

0

n

1

i

!

Decimal value of

∑

i

=

0

n

1

i

!

1

2

1

2.0000000000000000000

2

5

2

2.5000000000000000000

3

8

3

2.6666666666666666667

4

65

24

2.7083333333333333333

5

163

60

2.7166666666666666667

6

1957

720

2.7180555555555555556

7

685

252

2.7182539682539682540

8

109601

40320

2.7182787698412698413

9

98641

36288

2.7182815255731922399

10

9864101

3628800

2.7182818011463844797

11

13563139

4989600

2.7182818261984928652

12

260412269

95800320

2.7182818282861685639

13

8463398743

3113510400

2.7182818284467590023

14

47395032961

17435658240

2.7182818284582297479

15

888656868019

326918592000

2.7182818284589944643

16

56874039553217

20922789888000

2.7182818284590422591

17

7437374403113

2736057139200

2.7182818284590450705

18

17403456103284421

6402373705728000

2.7182818284590452267

19

82666416490601

30411275102208

2.7182818284590452349

20

6613313319248080001

2432902008176640000

2.7182818284590452353

21

69439789852104840011

25545471085854720000

2.7182818284590452354

22

611070150698522592097

224800145555521536000

2.7182818284590452354

23

1351405140967886501753

497154168055480320000

2.7182818284590452354

24

337310723185584470837549

124089680346647887872000

2.7182818284590452354

25

85351903640077042215979

31399210614030336000000

2.7182818284590452354

As the table shows, small values of

n

, are sufficient to give a good approximation of the

natural log base

e

, which at

n

=

21

is at 20 places undistinguishable from

e

.

Title

values of

∑

@

⁢

\slimits

⁢

@

⁢

@

⁢

@

i

=

0

n

⁢

1

i

!

for

0

<

n

<

26

Canonical name

ValuesOfdisplaystylesumi0nfrac1iFor0N26

Date of creation

2013-03-22 17:06:13

Last modified on

2013-03-22 17:06:13

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 33B99

Generated on Fri Feb 9 20:36:20 2018 by

LaTeXML
Barnes’ integral representation of the hypergeometric function

Barnes’ integral representation of the hypergeometric function

When

a

,

b

,

c

,

d

are

complex numbers

and

z

is a complex number
such that

-

π

<

arg

⁡

(

-

z

)

<

+

π

and

C

is a

contour

in the

complex

s

-plane which goes from

-

i

⁢

∞

to

+

i

⁢

∞

chosen such that the poles of

Γ

⁢

(

a

+

s

)

⁢

Γ

⁢

(

b

+

s

)

lie
to the left of

C

and the poles of

Γ

⁢

(

-

s

)

lie to the
right of

C

, then

∫

C

Γ

⁢

(

a

+

s

)

⁢

Γ

⁢

(

b

+

s

)

Γ

⁢

(

c

+

s

)

⁢

Γ

⁢

(

-

s

)

⁢

(

-

z

)

s

⁢

𝑑

s

=

2

⁢

π

⁢

i

⁢

Γ

⁢

(

a

)

⁢

Γ

⁢

(

b

)

Γ

⁢

(

c

)

⁢

F

⁢

(

a

,

b

;

c

;

z

)

Title

Barnes’

integral

representation of the

hypergeometric function

Canonical name

BarnesIntegralRepresentationOfTheHypergeometricFunction

Date of creation

2013-03-22 17:36:15

Last modified on

2013-03-22 17:36:15

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 33C05

Generated on Fri Feb 9 20:36:24 2018 by

LaTeXML
differential-difference equations for hypergeometric function

differential-difference equations for hypergeometric function

The

hypergeometric function

satisfies

several equations which
relate

derivatives

with respect to the

argument

z

to
shifting the

parameters

a

,

b

,

c

,

d

by unity (Here, the
prime denotes derivative with respect to

z

.):

z

⁢

F

′

⁢

(

a

,

b

;

c

;

z

)

+

a

⁢

F

⁢

(

a

,

b

;

c

;

z

)

=

F

⁢

(

a

+

1

,

b

;

c

;

z

)

z

⁢

F

′

⁢

(

a

,

b

;

c

;

z

)

+

b

⁢

F

⁢

(

a

,

b

;

c

;

z

)

=

F

⁢

(

a

,

b

+

1

;

c

;

z

)

z

⁢

F

′

⁢

(

a

,

b

;

c

;

z

)

+

(

c

-

1

)

⁢

F

⁢

(

a

,

b

;

c

;

z

)

=

F

⁢

(

a

,

b

;

c

-

1

;

z

)

(

1

-

z

)

⁢

z

⁢

F

′

⁢

(

a

,

b

;

c

;

z

)

=

(

c

-

a

)

⁢

F

⁢

(

a

-

1

,

b

;

c

;

z

)

+

(

a

-

c

+

b

⁢

z

)

⁢

F

⁢

(

a

,

b

;

c

;

z

)

(

1

-

z

)

⁢

z

⁢

F

′

⁢

(

a

,

b

;

c

;

z

)

=

(

c

-

b

)

⁢

F

⁢

(

a

,

b

-

1

;

c

;

z

)

+

(

b

-

c

+

a

⁢

z

)

⁢

F

⁢

(

a

,

b

;

c

;

z

)

(

1

-

z

)

⁢

z

⁢

F

′

⁢

(

a

,

b

;

c

;

z

)

=

z

⁢

(

c

-

a

)

⁢

(

c

-

b

)

⁢

F

⁢

(

a

,

b

;

c

+

1

;

z

)

+

z

⁢

c

⁢

(

a

+

b

-

c

)

⁢

F

⁢

(

a

,

b

;

c

;

z

)

These equations may readily be verified by differentiating the series
which defines the

hypergeometric equation

. By eliminating the derivatives
between these equations, one obtains the contiguity

relations

for the
hypergeometric function. By differentiating them once more and taking
suitable

linear combinations

, one may obtain the

differential equation

of the hypergeometric function.

Title

differential-difference equations for hypergeometric function

Canonical name

DifferentialdifferenceEquationsForHypergeometricFunction

Date of creation

2013-03-22 17:36:18

Last modified on

2013-03-22 17:36:18

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 33C05

Generated on Fri Feb 9 20:36:27 2018 by

LaTeXML
global characterization of hypergeometric function

global characterization of hypergeometric function

Riemann noted that the

hypergeometric function

can be characterized
by its global

properties

, without reference to

power series

,

differential
equations

, or any other sort of explicit

expression

. His

characterization

is conveniently restated in terms of sheaves:

Suppose that we have a sheaf of

holomorphic functions

over

ℂ

∖

{

0

,

1

}

which

satisfy

the following properties:

•

It is

closed under

analytic continuation

.

•

It is closed under taking

linear combinations

.

•

The

space of function

elements

over any open set is two dimensional.

•

There exists a neighborhood

D

0

such that

0

∈

D

)

, holomorphic
functions

ϕ

0

,

ψ

0

defined on

D

0

, and

complex numbers

α

0

,

β

0

such that, for an open set of

d

0

not containing

0

, it happens that

z

↦

z

α

0

⁢

ϕ

⁢

(

z

)

and

z

↦

z

β

0

⁢

ψ

⁢

(

z

)

belong to
our sheaf.

Then the sheaf consists of solutions to a

hypergeometric equation

, hence
the

function elements

are hypergeometric functions.

Title

global characterization of hypergeometric function

Canonical

name

GlobalCharacterizationOfHypergeometricFunction

Date of creation

2014-12-31 15:15:16

Last modified on

2014-12-31 15:15:16

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 33C05

Generated on Fri Feb 9 20:36:28 2018 by

LaTeXML
hypergeometric equation

hypergeometric equation

The

hypergeometric equation

is the following

linear ordinary differential equation

:

x

⁢

(

1

-

x

)

⁢

y

′′

+

(

c

-

(

a

+

b

+

1

)

⁢

x

)

⁢

y

′

-

a

⁢

b

⁢

y

=

0

(Here,

a

,

b

, and

c

are complex constants.)

The solutions of this equation may be expressed in terms of the

hypergeometric function

, hence the name.

The hypergeometric equation is a

Fuchsian differential equation

with singularities at

0

,

1

, and

∞

. By a suitable change of variables, any second order Fuchsian differential equation may be converted into a hypergeometric equation.

Title

hypergeometric equation

Canonical name

HypergeometricEquation

Date of creation

2013-03-22 14:45:53

Last modified on

2013-03-22 14:45:53

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 33C05

Generated on Fri Feb 9 20:36:30 2018 by

LaTeXML
hypergeometric function

hypergeometric function

Let

(

a

,

b

,

c

)

be a triple of complex numbers with

c

not belonging to the set of negative integers. For a complex number

w

and a non negative integer

n

, use

Pochhammer symbol

(

w

)

n

, to denote the

expression

:

(

w

)

n

=

w

⁢

(

w

+

1

)

⁢

…

⁢

(

w

+

n

-

1

)

.

The

Gauss hypergeometric function

,

F

1

2

, is then defined by the following power series expansion :

F

1

2

⁢

(

a

,

b

;

c

;

z

)

=

∑

n

=

0

∞

(

a

)

n

⁢

(

b

)

n

(

c

)

n

⁢

n

!

⁢

z

n

.

Title

hypergeometric function

Canonical

name

HypergeometricFunction

Date of creation

2013-03-22 14:27:48

Last modified on

2013-03-22 14:27:48

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 33C05

Related topic

TableOfMittagLefflerPartialFractionExpansions

Defines

Gauss hypergeometric function

Generated on Fri Feb 9 20:36:32 2018 by

LaTeXML
integral representation of the hypergeometric function

integral representation of the hypergeometric function

When

ℜ

⁡

c

>

ℜ

⁡

b

>

0

, one has the representation

F

⁢

(

a

,

b

;

c

;

z

)

=

Γ

⁢

(

c

)

Γ

⁢

(

b

)

⁢

Γ

⁢

(

c

-

b

)

⁢

∫

0

1

t

b

-

1

⁢

(

1

-

t

)

c

-

b

-

1

⁢

(

1

-

t

⁢

z

)

-

a

⁢

𝑑

t

Note that the conditions on

b

and

c

are necessary for the

integral

to be

convergent

at the

endpoints

0

and

1

. To see that this integral indeed equals the

hypergeometric function

, it suffices to consider the case

|

z

|

<

1

since both sides of the equation are

analytic functions

of

z

. (This follows from the

rigidity theorem for analytic functions

although some care is required because the

function

is multiply-valued.) With this

assumption

,

|

t

⁢

z

|

<

1

if

t

is a

real number

in the

interval

[

0

,

1

]

and hence,

(

1

-

t

⁢

z

)

-

a

may be

expanded

in a

power series

. Substituting this series in the right hand side of the

formula

above gives

Γ

⁢

(

c

)

Γ

⁢

(

b

)

⁢

Γ

⁢

(

c

-

b

)

⁢

∫

0

1

∑

k

=

0

∞

t

b

-

1

⁢

(

1

-

t

)

c

-

b

-

1

⁢

Γ

⁢

(

k

-

a

+

1

)

Γ

⁢

(

1

-

a

)

⁢

Γ

⁢

(

k

+

1

)

⁢

(

-

t

⁢

z

)

k

⁢

d

⁢

t

Since the series is

uniformly convergent

, it is permissible to integrate term-by-term. Interchanging integration and summation and pulling constants outside the integral sign, one obtains

Γ

⁢

(

c

)

Γ

⁢

(

b

)

⁢

Γ

⁢

(

c

-

b

)

⁢

∑

k

=

0

∞

Γ

⁢

(

k

-

a

+

1

)

Γ

⁢

(

1

-

a

)

⁢

Γ

⁢

(

k

+

1

)

⁢

(

-

z

)

k

⁢

∫

0

1

(

1

-

t

)

c

-

b

-

1

⁢

t

b

+

k

-

1

⁢

𝑑

t

The integrals appearing inside the sum are Euler beta functions. Expressing them in terms of

gamma functions

and simplifying, one sees that this integral indeed equals the hypergeometric function.

The hypergeometic function is multiply-valued. To obtain different branches of the hypergeometric function, one can vary the path of integration.

Title

integral representation of the hypergeometric function

Canonical name

IntegralRepresentationOfTheHypergeometricFunction

Date of creation

2013-03-22 14:35:14

Last modified on

2013-03-22 14:35:14

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 33C05

Generated on Fri Feb 9 20:36:34 2018 by

LaTeXML