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AppellF1

The AppellF1 function

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

AppellF1(a,b1,b2,c,z1,z2)

Parameters

a

-

algebraic expression

b1

-

algebraic expression

b2

-

algebraic expression

c

-

algebraic expression

z1

-

algebraic expression

z2

-

algebraic expression

Description

• 

As is the case of all the four multi-parameter Appell functions, AppellF1, is a doubly hypergeometric function that includes as particular cases the 2F1 hypergeometric and some cases of the MeijerG function, and with them most of the known functions of mathematical physics. Among other situations, AppellF1 appears in the solution to differential equations in general relativity, quantum mechanics, and molecular and atomic physics.

  

Initialization: Set the display of special functions in output to typeset mathematical notation (textbook notation):

Typesetting:-EnableTypesetRuleTypesetting:-SpecialFunctionRules:

  

The definition of the AppellF1 series and the corresponding domain of convergence can be seen through the FunctionAdvisor

FunctionAdvisordefinition,AppellF1

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=_k1=0_k2=0a_k1+_k2b__1_k1b__2_k2z__1_k1z__2_k2c_k1+_k2_k1!_k2!&comma;z__1<1z__2<1

(1)
  

A distinction is made between the AppellF1 doubly hypergeometric series, with the restricted domain of convergence shown above, and the AppellF1 function, that coincides with the series in its domain of convergence but also extends it analytically to the whole complex plane.

  

From the definition above, by swapping the AppellF1 variables subscripted with the numbers 1 and 2, the function remains the same; hence

FunctionAdvisorsymmetries&comma;AppellF1

F1a&comma;b__2&comma;b__1&comma;c&comma;z__2&comma;z__1=F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

(2)
  

From the series' definition, AppellF1 is singular (division by zero) when the c parameter entering the pochhammer function in the denominator of the series is a non-positive integer because the pochhammer function will be equal to zero when the summation index of the series is bigger than the absolute value of c.

  

For an analogous reason, when the a and/or both b1 and b2 parameters entering the pochhammer functions in the numerator of the series are non-positive integers, the series will truncate and AppellF1 will be polynomial. As is the case of the hypergeometric function, when the pochhammers in both the numerator and the denominator have non-positive integer arguments, AppellF1 is polynomial if the absolute value of the non-positive integers in the pochhammers of the numerator are smaller than or equal to the absolute value of the non-positive integer (parameter c) in the pochhammer in the denominator, and singular otherwise. Consult the FunctionAdvisor for comprehensive information on the combinations of all these conditions. For example, the singular cases happen when any of the following conditions hold

FunctionAdvisorsingularities&comma;AppellF1

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2&comma;c::&apos;nonposint&apos;a::&apos;nonposint&apos;b__1::¬&apos;nonposint&apos;a<cc::&apos;nonposint&apos;a::&apos;nonposint&apos;b__2::¬&apos;nonposint&apos;a<cc::&apos;nonposint&apos;a::&apos;nonposint&apos;b__1::&apos;nonposint&apos;b__2::&apos;nonposint&apos;a<cb__1+b__2<cc::&apos;nonposint&apos;a::¬&apos;nonposint&apos;b__1::¬&apos;nonposint&apos;c::&apos;nonposint&apos;a::¬&apos;nonposint&apos;b__2::¬&apos;nonposint&apos;c::&apos;nonposint&apos;a::¬&apos;nonposint&apos;b__1::&apos;nonposint&apos;b__2::&apos;nonposint&apos;b__1+b__2<c

(3)
  

The AppellF1 series is analytically extended to the AppellF1 function defined over the whole complex plane using identities and mainly by integral representations in terms of Eulerian integrals:

FunctionAdvisorintegral_form&comma;AppellF1

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=Γc011u1+b__1uz__1+1c+aF12a,b__2;cb__1;z__2uuc+b__1+1&DifferentialD;uΓb__1Γcb__11z__1c+a+b__1&comma;0<b__10<c+b__1,F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=Γc011ub__21z__2u+1c+aF12a,b__1;cb__2;uz__1uc+b__2+1&DifferentialD;uΓb__2Γcb__21z__2c+a+b__2&comma;0<b__20<c+b__2,F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=Γc01ua11uc+a+1uz__1+1b__1z__2u+1b__2&DifferentialD;uΓaΓca&comma;0<a0<c+a,F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=Γc0101vu1+b__1vb__211uvc+b__1+b__2+1uz__1vz__2+1a&DifferentialD;u&DifferentialD;vΓcb__1b__2Γb__1Γb__2&comma;0<b__10<b__20<c+b__1+b__2

(4)
  

These integral representations are also the starting point for the derivation of many of the identities known for AppellF1.

  

AppellF1 also satisfies a linear system of partial differential equations of second order

FunctionAdvisorDE&comma;AppellF1

fa&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2&comma;2z__12fa&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__22z__1z__2fa&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2z__1+ab__11z__1+cz__1fa&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2z__11+z__1b__1z__2z__2fa&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2z__11+z__1fa&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2ab__1z__11+z__1&comma;2z__1z__2fa&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__22z__22fa&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2z__1b__2z__1fa&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2z__21+ab__21z__2+cz__2fa&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2z__1z__21fa&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2ab__2z__1z__21

(5)

Examples

  

Initialization: Set the display of special functions in output to typeset mathematical notation (textbook notation):

Typesetting:-EnableTypesetRuleTypesetting:-SpecialFunctionRules&colon;

The conditions for both the singular and the polynomial cases can also be seen from the AppellF1. For example, the six polynomial cases of AppellF1 are

AppellF1:-SpecialValues:-Polynomial

6,a&comma;b1&comma;b2&comma;c&comma;z1&comma;z2a::&apos;nonposint&apos;&comma;c::¬&apos;nonposint&apos;&comma;a::&apos;nonposint&apos;&comma;c::&apos;nonposint&apos;&comma;ca&comma;b1::&apos;nonposint&apos;&comma;b2::&apos;nonposint&apos;&comma;c::¬&apos;nonposint&apos;&comma;b1::&apos;nonposint&apos;&comma;b2::&apos;nonposint&apos;&comma;c::&apos;nonposint&apos;&comma;cb1+b2&comma;b1::&apos;nonposint&apos;&comma;c::¬&apos;nonposint&apos;&comma;b2::&apos;nonposint&apos;&comma;c::¬&apos;nonposint&apos;

(6)

Likewise, the conditions for the singular cases of AppellF1 can be seen either using the FunctionAdvisor or entering AppellF1:-Singularities(), so with no arguments.

For particular values of its parameters, AppellF1 is related to the hypergeometric and elliptic functions. These hypergeometric cases are returned automatically. For example, for z1=1,

%AppellF1=AppellF1a&comma;b__1&comma;b__2&comma;c&comma;1&comma;z__2

%AppellF1a&comma;b__1&comma;b__2&comma;c&comma;1&comma;z__2=F12a,b__1;c;1F12a,b__2;cb__1;z__2

(7)

This formula analytically extends to the whole complex plane the AppellF1 series when any of z1=1 or z2=1 (the latter using the symmetry of AppellF1 - see the beginning of the Description section).

To see all the hypergeometric cases, enter

FunctionAdvisorspecialize&comma;AppellF1&comma;hypergeom

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a,b__2;c;z__2&comma;z__1=0b__1=0,F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a,b__1;c;z__1&comma;z__2=0b__2=0,F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a,b__1;c;1F12a,b__2;cb__1;z__2&comma;z__1=1,F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a,b__2;c;1F12a,b__1;cb__2;z__1&comma;z__2=1,F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a,b__1+b__2;c;z__1&comma;z__1=z__2,F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F23b__1,a2,a2+12;c2,c2+12;z__12&comma;z__1=z__2b__1=b__2,F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F01b__1;;z__1F01b__2;;z__2&comma;c=aa0,F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__1F121,a;c;z__1z__2F121,a;c;z__2z__2+z__1&comma;b__1=1b__2=1z__1z__2

(8)

Other special values of AppellF1 can be seen using FunctionAdvisor(special_values, AppellF1).

By requesting the sum form of AppellF1, besides its double power series definition, we also see the particular form the series takes when one of the summations is performed and the result expressed in terms of 2F1 hypergeometric functions:

FunctionAdvisorsum_form&comma;AppellF1

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=m=0n=0am+nb__1mb__2nz__1mz__2ncm+nm!n!&comma;z__1<1z__2<1,F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=k=0akb__1kF12a+k,b__2;c+k;z__2z__1kckk!&comma;z__1<1,F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=k=0akb__2kF12a+k,b__1;c+k;z__1z__2kckk!&comma;z__2<1

(9)

As indicated in the formulas above, for AppellF1 (also for AppellF3) the domain of convergence of the single sum with hypergeometric coefficients is larger than the domain of convergence of the double series, because the hypergeometric coefficient in the single sum - say the one in z2 - analytically extends the series with regards to the other variable - say z1 - entering the hypergeometric coefficient. Hence, for AppellF1 (also for AppellF3), the case where one of the two variables, z1 or z2, is equal to 1, is convergent only when the corresponding hypergeometric coefficient in the single sum form is convergent. For instance, the convergent case at z1&equals;1 requires that 0<Reca b1.

AppellF1 admits identities analogous to Euler identities for the hypergeometric function. These Euler-type identities, as well as contiguity identities, are visible using the FunctionAdvisor with the option identities, or directly from the function. For example,

AppellF1a&comma; b__1&comma; b__2&comma; c&comma; z__1&comma; z__2 &equals; AppellF1:-TransformationsEuler1a&comma; b__1&comma; b__2&comma; c&comma; z__1&comma; z__2

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1z__1b__11z__2b__2F1ca&comma;b__1&comma;b__2&comma;c&comma;z__1z__11&comma;z__21+z__2

(10)

Among other situations, this identity is useful when both z1 and z2 have absolute values larger than 1 but one of the arguments in the same position of AppellF1 on the right-hand side has absolute value smaller than 1.

A contiguity transformation for AppellF1

AppellF1a&comma; b__1&comma; b__2&comma; c&comma; z__1&comma; z__2 &equals; AppellF1:-TransformationsContiguity1a&comma; b__1&comma; b__2&comma; c&comma; z__1&comma; z__2

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=c1F1a1&comma;b__1&comma;b__21&comma;c1&comma;z__1&comma;z__2F1a1&comma;b__11&comma;b__2&comma;c1&comma;z__1&comma;z__2z__2+z__1a1

(11)

The contiguity transformations available in this way are

indicesAppellF1:-TransformationsContiguity

1,2,3,4,5,6,7,9,8,10

(12)

By using differential algebra techniques, the PDE system satisfied by AppellF1 can be transformed into an equivalent PDE system where one of the equations is a linear ODE in z2 parametrized by z1. In the case of AppellF1 this linear ODE is of third order and can be computed as follows

F1z__1&comma;z__2 &equals; AppellF1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

F1z__1&comma;z__2=F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

(13)

simplifyop1&comma; 2&comma; PDEtools:-casesplitPDEtools:-dpolyform&comma; no_Fn&comma; lex

3z__23F1z__1&comma;z__2=a+2b__2+4z__22+a+b__1b__23z__1cb__22z__2+z__1cb__1+12z__22F1z__1&comma;z__2+2a+b__2+2z__2+a+b__11z__1cz__2F1z__1&comma;z__2+F1z__1&comma;z__2ab__2b__2+11+z__2z__2+z__1z__2

(14)

This linear ODE has four regular singularities, one of which is located at z1

DEtoolssingularitiessubsF1z__1&comma;z__2=F1z__2&comma;

regular=0&comma;1&comma;z__1&comma;,irregular=

(15)

You can also see a general presentation of AppellF1, organized into sections and including plots, using the FunctionAdvisor

FunctionAdvisorAppellF1

AppellF1

describe

AppellF1=Appell 2-variable hypergeometric function F1

definition

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=_k1=0_k2=0a_k1+_k2b__1_k1b__2_k2z__1_k1z__2_k2c_k1+_k2_k1!_k2!

z__1<1z__2<1

classify function

Appell

symmetries

F1a&comma;b__2&comma;b__1&comma;c&comma;z__2&comma;z__1=F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

plot

singularities

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

c::Typesetting:-_Hold&apos;nonposint&apos;a::Typesetting:-_Hold&apos;nonposint&apos;b__1::¬Typesetting:-_Hold&apos;nonposint&apos;a<cc::Typesetting:-_Hold&apos;nonposint&apos;a::Typesetting:-_Hold&apos;nonposint&apos;b__2::¬Typesetting:-_Hold&apos;nonposint&apos;a<cc::Typesetting:-_Hold&apos;nonposint&apos;a::Typesetting:-_Hold&apos;nonposint&apos;b__1::Typesetting:-_Hold&apos;nonposint&apos;b__2::Typesetting:-_Hold&apos;nonposint&apos;a<cb__1+b__2<cc::Typesetting:-_Hold&apos;nonposint&apos;a::¬Typesetting:-_Hold&apos;nonposint&apos;b__1::¬Typesetting:-_Hold&apos;nonposint&apos;c::Typesetting:-_Hold&apos;nonposint&apos;a::¬Typesetting:-_Hold&apos;nonposint&apos;b__2::¬Typesetting:-_Hold&apos;nonposint&apos;c::Typesetting:-_Hold&apos;nonposint&apos;a::¬Typesetting:-_Hold&apos;nonposint&apos;b__1::Typesetting:-_Hold&apos;nonposint&apos;b__2::Typesetting:-_Hold&apos;nonposint&apos;b__1+b__2<c

branch points

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

a::¬Typesetting:-_Hold&apos;nonposint&apos;b__1::¬Typesetting:-_Hold&apos;nonposint&apos;z__11&comma;+Ia::¬Typesetting:-_Hold&apos;nonposint&apos;b__2::¬Typesetting:-_Hold&apos;nonposint&apos;z__21&comma;+I

branch cuts

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

a::¬Typesetting:-_Hold&apos;nonposint&apos;b__1::¬Typesetting:-_Hold&apos;nonposint&apos;1<z__1a::¬Typesetting:-_Hold&apos;nonposint&apos;b__2::¬Typesetting:-_Hold&apos;nonposint&apos;1<z__2

special values

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1

z__1=0z__2=0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1

a=0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1

b__1=0b__2=0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a,b__2;c;z__2

z__1=0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a,b__2;c;z__2

b__1=0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a,b__1;c;z__1

z__2=0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a,b__1;c;z__1

b__2=0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a,b__1;c;1F12a,b__2;cb__1;z__2

z__1=1

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a,b__2;c;1F12a,b__1;cb__2;z__1

z__2=1

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a,b__1+b__2;c;z__1

z__1=z__2

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F23b__1,a2,a2+12;c2,c2+12;z__12

z__1=z__2b__1=b__2

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F01b__1;;z__1F01b__2;;z__2

c=aa0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__1F121,a;c;z__1z__2F121,a;c;z__2z__2+z__1

b__1=1b__2=1z__1z__2

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=arctanhz__1z__1arctanhz__2z__2z__2+z__1

a=12b__1=1b__2=1c=32

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=3arctanhz__1z__2z__1z__11z__1z__2z__1z__1arcsinz__1z__1z__2

a=32b__1=12b__2=1c=52

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=2aarctanhz__11+z__2z__11z__1z__212+a1+z__2z__1_k2=032+a−1_k212+a_k2+12z__1+z__2z__2_k2+1_k1=0_k2+1_k2+1_k1z__1_k1_k4=0_k12+121_k112_k412_k42z__1z__2z__1_k112_k4z__22_k42arcsinz__1+1z__1z__1_k3=1_k4_k31!12z__12_k3112_k3_k4!z__12_k4+_k4=01+_k1222_k4+1_k112_k4+1_k4!1z__1_k4+12z__112_k42z__1z__2z__1_k122_k4z__22_k4+1_k3=0_k412z__12_k322_k31z__1_k3z__1_k3_k3!32_k4_k32z__1+z__2_k1212+az__112+az__1

12+a::Typesetting:-_Hold&apos;nonnegint&apos;b__1=12b__2=1c1a=0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=3z__1z__12arctanhz__21+z__1z__2z__2+arctanhz__11+z__1z__2+z__121z__1z__1z__2+z__12

a=32b__1=2b__2=1c=52

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=5arctanhz__11+z__123z__12+6z__1z__2z__22+z__15z__138arctanhz__21+z__12z__1z__2+z__22+z__1z__22+z__23z__121+2z__281+z__12z__132z__2+z__13

a=52b__1=3b__2=1c=72

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=2c2−152+c1z__232+cz__212cz__2z__11+z__2arctanhz__2z__11+z__2+−132+c1z__1c12_k1=052+c_k1+12_k12c+3z__1_k12+2c2c5_k1!z__1z__2z__11+z__22_k1+z__1+z__2z__11+z__22_k1arctanhz__1_k2=0_k1−1_k2z__1z__2z__11+z__2_k2_k11+z__1+z__2z__11+z__2_k2_k11_k3=0_k22_k22_k31z__11_k2+_k3z__1_k22_k3_k2_k3!_k22_k3!_k3!_k2+1c52_k1!32+c!1z__2

a=1b__1=12b__2=132+c::Typesetting:-_Hold&apos;nonnegint&apos;

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1b__1=0b__2=0a=0_k2=0ac_k1=0ac_k2a+c_k1+_k2b__1_k2b__2_k1z__11+z__1_k2z__21+z__2_k1c_k1+_k2_k2!_k1!1z__1b__11z__2b__2otherwise

z__11z__21ac::Typesetting:-_Hold&apos;nonnegint&apos;

identities

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1z__1b__11z__2b__2F1ca&comma;b__1&comma;b__2&comma;c&comma;z__11+z__1&comma;z__21+z__2

z__11z__21a::Typesetting:-_Hold&apos;nonposint&apos;b__1::Typesetting:-_Hold&apos;nonposint&apos;b__2::Typesetting:-_Hold&apos;nonposint&apos;¬1<z__11<z__2

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1z__1aF1a&comma;cb__1b__2&comma;b__2&comma;c&comma;z__11+z__1&comma;z__1z__21+z__1

z__11a::Typesetting:-_Hold&apos;nonposint&apos;b__1::Typesetting:-_Hold&apos;nonposint&apos;b__2::Typesetting:-_Hold&apos;nonposint&apos;¬1<z__11<z__2

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1z__1cab__11z__2b__2F1ca&comma;cb__1b__2&comma;b__2&comma;c&comma;z__1&comma;z__2z__11+z__2

z__21z__11

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1+cF11+a&comma;b__1&comma;b__21&comma;1+c&comma;z__1&comma;z__2F11+a&comma;b__11&comma;b__2&comma;1+c&comma;z__1&comma;z__2z__1z__21+a

z__1z__2a1c1

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=anF1n+a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2ac+1nk=1n−1knk1ckF1a&comma;b__1&comma;b__2&comma;ck&comma;z__1&comma;z__2ac+1k

c1ac+1::¬Typesetting:-_Hold&apos;nonposint&apos;nac+1

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=aF1a+1&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2b__1F1a&comma;b__1+1&comma;b__2&comma;c&comma;z__1&comma;z__2b__2F1a&comma;b__1&comma;b__2+1&comma;c&comma;z__1&comma;z__2b__1b__2+a

ab__1+b__2

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=caz__1F1a&comma;b__1&comma;b__2&comma;c+1&comma;z__1&comma;z__2cF1a&comma;b__11&comma;b__2&comma;c&comma;z__1&comma;z__2c1+z__1

c0z__11

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=caz__2F1a&comma;b__1&comma;b__2&comma;c+1&comma;z__1&comma;z__2cF1a&comma;b__1&comma;b__21&comma;c&comma;z__1&comma;z__2c1+z__2

c0z__21

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1n+a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2b__1z__1k=1nF1a+k&comma;b__1+1&comma;b__2&comma;c+1&comma;z__1&comma;z__2cb__2z__2k=1nF1a+k&comma;b__1&comma;b__2+1&comma;c+1&comma;z__1&comma;z__2c

c0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1a&comma;b__1+n&comma;b__2&comma;c&comma;z__1&comma;z__2az__1k=1nF1a+1&comma;b__1+k&comma;b__2&comma;c+1&comma;z__1&comma;z__2c

c0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1an&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2+b__2z__2k=0n1F1ak&comma;b__1&comma;b__2+1&comma;c+1&comma;z__1&comma;z__2c+b__1z__1k=0n1F1ak&comma;b__1+1&comma;b__2&comma;c+1&comma;z__1&comma;z__2c

c0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1a&comma;b__1n&comma;b__2&comma;c&comma;z__1&comma;z__2+az__1k=0n1F1a+1&comma;b__1k&comma;b__2&comma;c+1&comma;z__1&comma;z__2c

c0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__1z__2b__2F2b__1+b__2&comma;a&comma;b__2&comma;c&comma;b__1+b__2&comma;z__1&comma;1z__1z__2

z__10z__20

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=11z__1b__1F3a&comma;b__1&comma;b__2&comma;ca&comma;c&comma;z__2&comma;z__11+z__1

z__11

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=11z__2b__2F3a&comma;b__2&comma;b__1&comma;ca&comma;c&comma;z__1&comma;z__21+z__2

z__21

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=Physics:−Library:−Addak1+k2F3b__1&comma;b__2&comma;k1&comma;k2&comma;c&comma;z__1&comma;z__2k1!k2!&comma;k1+k2a−1a

a::Typesetting:-_Hold&apos;nonposint&apos;

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__1z__2b__2F4a&comma;b__1+b__2&comma;c&comma;b__1+b__2&comma;z__12z__2&comma;z__1z__21+z__1z__2

1c=0ab__2=0

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__1z__2b__2F4b__12+b__22&comma;12+b__12+b__22&comma;12+a&comma;b__2+12&comma;z__12z__221+z__2z__1z__22&comma;z__1z__221+z__1z__2z__1212z__12+z__12z__2b__1+b__2

c2a=0b__2+b__1=012z__12+z__12z__20

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ab__112=0b__2+b__1=0z__1z__22+z__1z__221

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__1z__2b__21+4z__1+81z__1+z__128z__1+8z__122b__1+2b__2F4b__2&comma;b__1+b__2&comma;b__1+1&comma;b__1+b__2&comma;4z__1+81z__1+z__128z__1+8z__12&comma;24z__1+81z__1+z__128z__1+8z__12z__12+z__12z__121z__12z__1z__2z__12z__2

ab__112=0c2a=0z__112+z__121

sum form

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=m=0n=0am+nb__1mb__2nz__1mz__2ncm+nm!n!

z__1<1z__2<1

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=k=0akb__1kF12a+k,b__2;c+k;z__2z__1kckk!

z__1<1

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=k=0akb__2kF12a+k,b__1;c+k;z__1z__2kckk!

z__2<1

series

seriesF1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2&comma;z__1&comma;4=F12a,b__2;c;z__2+ab__1F12b__2,a+1;c+1;z__2cz__1+12ab__1a+1b__1+1F12b__2,a+2;c+2;z__2cc+1z__12+16ab__1a+1b__1+1a+2b__1+2F12b__2,a+3;c+3;z__2cc+1c+2z__13+Oz__14

seriesF1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2&comma;z__2&comma;4=F12a,b__1;c;z__1+ab__2F12b__1,a+1;c+1;z__1cz__2+12ab__2a+1b__2+1F12b__1,a+2;c+2;z__1cc+1z__22+16ab__2a+1b__2+1a+2b__2+2F12b__1,a+3;c+3;z__1cc+1c+2z__23+Oz__24

integral form

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=Γc011u1+b__1uz__1+1c+aF12a,b__2;cb__1;z__2uuc+b__1+1&DifferentialD;uΓb__1Γcb__11z__1c+a+b__1

0<b__10<c+b__1

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=Γc011ub__21z__2u+1c+aF12a,b__1;cb__2;uz__1uc+b__2+1&DifferentialD;uΓb__2Γcb__21z__2c+a+b__2

0<b__20<c+b__2

F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=Γc01ua11uc+a+1uz__1+1b__1z__2u+1b__2&DifferentialD;uΓaΓca

0<a0<c+a

differentiation rule

z__1F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=ab__1F1a+1&comma;b__1+1&comma;b__2&comma;c+1&comma;z__1&comma;z__2c

nz__1nF1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=anb__1nF1n+a&comma;n+b__1&comma;b__2&comma;n+c&comma;z__1&comma;z__2cn

z__2F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=ab__2F1a+1&comma;b__1&comma;b__2+1&comma;c+1&comma;z__1&comma;z__2c

nz__2nF1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=anb__2nF1n+a&comma;b__1&comma;n+b__2&comma;n+c&comma;z__1&comma;z__2cn

DE

fa&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1a&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

2z__12fa&comma;b__1&comma;b__2&comma;c&