AppellF3 - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


AppellF3

The AppellF3 function

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

AppellF3(a1,a2,b1,b2,c,z1,z2)

Parameters

a1

-

algebraic expression

a2

-

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, AppellF3, 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, AppellF3 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 AppellF3 series and the corresponding domain of convergence can be seen through the FunctionAdvisor

FunctionAdvisordefinition,AppellF3

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=_k1=0_k2=0a__1_k1a__2_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 AppellF3 doubly hypergeometric series, with the restricted domain of convergence shown above, and the AppellF3 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 AppellF3 variables subscripted with the numbers 1 and 2, the function remains the same; hence

FunctionAdvisorsymmetries&comma;AppellF3

F3a__2&comma;a__1&comma;b__2&comma;b__1&comma;c&comma;z__2&comma;z__1=F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2&comma;F3b__1&comma;a__2&comma;a__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2&comma;F3a__1&comma;b__2&comma;b__1&comma;a__2&comma;c&comma;z__1&comma;z__2=F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

(2)
  

Note the existence of other symmetries, also visible in the double sum definition.

  

From the series' definition, AppellF3 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 a1 and/or a2 and/or b1 and/or b2 parameters entering the pochhammer functions in the numerator of the series are non-positive integers, the series will truncate and AppellF3 will be polynomial in one of the two of z1,z2. As is the case of the hypergeometric function, when the pochhammers in both the numerator and the denominator have non-positive integer arguments, AppellF3 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;AppellF3

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2&comma;c::0&comma;a__1::¬0&comma;b__1::¬0&comma;c::0&comma;a__2::¬0&comma;b__2::¬0&comma;c::0&comma;a__1::¬0&comma;a__2::0&comma;b__1::0&comma;b__2::¬0&comma;b__1+a__2<cc::0&comma;a__1::¬0&comma;a__2::¬0&comma;b__1::0&comma;b__2::0&comma;b__1+b__2<cc::0&comma;a__1::¬0&comma;a__2::0&comma;b__1::0&comma;b__2::0&comma;b__1+a__2<cb__1+b__2<cc::0&comma;a__1::0&comma;a__2::0&comma;b__1::¬0&comma;b__2::¬0&comma;a__1+a__2<cc::0&comma;a__1::0&comma;a__2::¬0&comma;b__1::¬0&comma;b__2::0&comma;a__1+b__2<cc::0&comma;a__1::0&comma;a__2::0&comma;b__1::¬0&comma;b__2::0&comma;a__1+a__2<ca__1+b__2<cc::0&comma;a__1::0&comma;a__2::0&comma;b__1::0&comma;b__2::0&comma;a__1+a__2<ca__1+b__2<cb__1+a__2<cb__1+b__2<cc::0&comma;a__1::0&comma;a__2::¬0&comma;b__1::0&comma;b__2::0&comma;a__1+b__2<cb__1+b__2<cc::0&comma;a__1::0&comma;a__2::0&comma;b__1::0&comma;b__2::¬0&comma;a__1+a__2<cb__1+a__2<c

(3)
  

The AppellF3 series is analytically extended to the AppellF3 function defined over the whole complex plane using identities and mainly by integral representations in terms of Eulerian integrals:

FunctionAdvisorintegral_form&comma;AppellF3

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=Γc011u1+b__1F12a__2,b__2;cb__1;z__2uuc+b__1+11+u1z__1a__1&DifferentialD;uΓb__1Γcb__1&comma;0<b__10<c0<c+b__1,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=Γc011ub__21F12a__1,b__1;cb__2;z__1uuc+b__2+11+u1z__2a__2&DifferentialD;uΓb__2Γcb__2&comma;0<b__20<c0<c+b__2,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=Γc01uρ11ucρ1F12a__1,b__1;ρ;z__1uF12a__2,b__2;cρ;u1z__2&DifferentialD;uΓρΓcρ&comma;0<c,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=Γc0101vu1+b__1vb__211uvc+b__1+b__2+1z__1u+1a__1vz__2+1a__2&DifferentialD;u&DifferentialD;vΓb__1Γb__2Γcb__1b__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 AppellF3.

  

AppellF3 also satisfies a linear system of partial differential equations of second order

FunctionAdvisorDE&comma;AppellF3

fa__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2&comma;2z__12fa__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__22z__1z__2fa__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2z__1z__11+a__1b__11z__1+cz__1fa__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2z__1z__11a__1b__1fa__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2z__1z__11&comma;2z__1z__2fa__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__2z__212z__22fa__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2z__1+a__2+b__2+1z__2cz__2fa__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2z__1+a__2b__2fa__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2z__1

(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 AppellF3. For example, the twelve polynomial cases of AppellF3 are

AppellF3:-SpecialValues:-Polynomial

12,a1&comma;a2&comma;b1&comma;b2&comma;c&comma;z1&comma;z2a1::0&comma;&comma;a2::0&comma;&comma;c::¬0&comma;&comma;a1::0&comma;&comma;b2::0&comma;&comma;c::¬0&comma;&comma;b1::0&comma;&comma;a2::0&comma;&comma;c::¬0&comma;&comma;b1::0&comma;&comma;b2::0&comma;&comma;c::¬0&comma;&comma;a1::0&comma;&comma;a2::0&comma;&comma;c::0&comma;&comma;ca1+a2&comma;a1::0&comma;&comma;b2::0&comma;&comma;c::0&comma;&comma;ca1+b2&comma;b1::0&comma;&comma;a2::0&comma;&comma;c::0&comma;&comma;cb1+a2&comma;b1::0&comma;&comma;b2::0&comma;&comma;c::0&comma;&comma;cb1+b2&comma;a1::0&comma;&comma;c::¬0&comma;&comma;a2::0&comma;&comma;c::¬0&comma;&comma;b1::0&comma;&comma;c::¬0&comma;&comma;b2::0&comma;&comma;c::¬0&comma;

(6)

Likewise, the conditions for the singular cases of AppellF3 can be seen either using the FunctionAdvisor or entering AppellF3:-Singularities(), so with no arguments.

For particular values of its parameters, AppellF3 is related to the hypergeometric function. These hypergeometric cases are returned automatically. For example, for z1=1,

%AppellF3=AppellF3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;1&comma;z__2

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;1&comma;z__2=F12a__1,b__1;c;1F23a__2,b__2,ca__1b__1;cb__1,ca__1;z__2

(7)

This formula analytically extends to the whole complex plane the AppellF3 series when any of z1=1 or z2=1 (the latter using the symmetry of AppellF3 - see the beginning of the Description section).

To see all the hypergeometric cases, enter

FunctionAdvisorspecialize&comma;AppellF3&comma;hypergeom

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a__2,b__2;c;z__2&comma;z__1=0a__1=0b__1=0,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a__1,b__1;c;z__1&comma;z__2=0a__2=0b__2=0,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a__1,b__1;c;1F23a__2,b__2,ca__1b__1;cb__1,ca__1;z__2&comma;z__1=1,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a__2,b__2;c;1F23a__1,b__1,ca__2b__2;cb__2,ca__2;z__1&comma;z__2=1,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__1F121,a__1;a__1+a__2;z__1z__2F121,a__2;a__1+a__2;z__2z__21z__1z__2&comma;b__1=1b__2=1c=a__1+a__2z__1z__2+z__1+z__20,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__1F121,b__1;b__1+a__2;z__1z__2F121,a__2;b__1+a__2;z__2z__21z__1z__2&comma;a__1=1b__2=1c=b__1+a__2z__1z__2+z__1+z__20,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__1F121,a__1;a__1+b__2;z__1z__2F121,b__2;a__1+b__2;z__2z__21z__1z__2&comma;b__1=1a__2=1c=a__1+b__2z__1z__2+z__1+z__20,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1z__1b__2F12a__1+b__2,b__1+b__2;a__1+a__2+b__1+b__2;z__1&comma;c=a__1+a__2+b__1+b__2z__11z__2=z__1z__11c=a__1+a__2+b__1+b__2z__21z__1=z__2z__21,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1z__1c1F12c2+a__12a__22,c2a__12a__22+12;c;4z__1z__1112z__11+a__2&comma;z__2=z__12z__112z__11z__11b__1=1a__1b__2=1a__2z__1=z__22z__212z__21z__21b__1=1a__1b__2=1a__2z__2=z__12z__112z__11z__11a__1=1b__1b__2=1a__2z__2=z__12z__112z__11z__11b__1=1a__1a__2=1b__2

(8)

Other special values of AppellF3 can be seen using FunctionAdvisor(special_values, AppellF3).

By requesting the sum form of AppellF3, 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;AppellF3

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=m=0n=0a__1ma__2nb__1mb__2nz__1mz__2ncm+nm!n!&comma;z__1<1z__2<1,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=k=0a__1kb__1kF12a__2,b__2;c+k;z__2z__1kckk!&comma;z__1<1,F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=k=0a__2kb__2kF12a__1,b__1;c+k;z__1z__2kckk!&comma;z__2<1

(9)

As indicated in the formulas above, for AppellF3 (also for AppellF1) 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 AppellF3 (also for AppellF1), 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<Reca1b1.

AppellF3 is the only one of the four Appell functions that does not admit identities analogous to the Euler identities for the hypergeometric function.

A contiguity transformation for AppellF3

AppellF3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2 &equals; AppellF3:-TransformationsContiguity1a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F3a__1&comma;a__2&comma;b__1&comma;b__2+n&comma;c&comma;z__1&comma;z__2k=1nnka__2kz__2kF3a__1&comma;b__2+k&comma;b__1&comma;k+a__2&comma;k+c&comma;z__1&comma;z__2ck&comma;z__21c::¬0&comma;a__2::0&comma;c<a__2nc

(10)

The contiguity transformations available in this way are

indicesAppellF3:-TransformationsContiguity

1,2,3,4,5,6

(11)

By using differential algebra techniques, the PDE system satisfied by AppellF3 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 AppellF3 this linear ODE is of fourth order and can be computed as follows

F3z__1&comma;z__2 &equals; AppellF3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

F3z__1&comma;z__2=F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

(12)

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

4z__24F3z__1&comma;z__2=2z__2z__11a__2+b__2+4z__22+ca__12+a__2b__12+b__2+112z__1c2a__22b__2252z__2ca__12b__12+32z__13z__23F3z__1&comma;z__2+a__22+4b__2+9a__2+b__22+9b__2+14z__11z__22+2ca__1b__1+2b__2+5a__2+2ca__1b__1+5b__2+6c3a__13b__1+11z__1+cb__22a__2+c2b__23c4z__2z__1cb__1+1ca__1+12z__22F3z__1&comma;z__2+2z__11a__2+b__2+2z__2+ca__12b__12+12z__1c2z__2F3z__1&comma;z__2a__2b__2F3z__1&comma;z__2z__112b__2+1a__2+1z__22z__21z__11z__2z__1

(13)

This linear ODE has four regular singularities, one of which depends on z1

DEtoolssingularitiessubsF3z__1&comma;z__2=F3z__2&comma;

regular=0&comma;1&comma;&comma;z__1z__11,irregular=

(14)

You can also see a general presentation of AppellF3, organized into sections and including plots, using the FunctionAdvisor

FunctionAdvisorAppellF3

AppellF3

describe

AppellF3=Appell 2-variable hypergeometric function F3

definition

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=_k1=0_k2=0a__1_k1a__2_k2b__1_k1b__2_k2z__1_k1z__2_k2c_k1+_k2_k1!_k2!

z__1<1z__2<1

classify function

Appell

symmetries

F3a__2&comma;a__1&comma;b__2&comma;b__1&comma;c&comma;z__2&comma;z__1=F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

F3b__1&comma;a__2&comma;a__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

F3a__1&comma;b__2&comma;b__1&comma;a__2&comma;c&comma;z__1&comma;z__2=F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

plot

singularities

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

c::0&comma;a__1::¬0&comma;b__1::¬0&comma;c::0&comma;a__2::¬0&comma;b__2::¬0&comma;c::0&comma;a__1::¬0&comma;a__2::0&comma;b__1::0&comma;b__2::¬0&comma;b__1+a__2<cc::0&comma;a__1::¬0&comma;a__2::¬0&comma;b__1::0&comma;b__2::0&comma;b__1+b__2<cc::0&comma;a__1::¬0&comma;a__2::0&comma;b__1::0&comma;b__2::0&comma;b__1+a__2<cb__1+b__2<cc::0&comma;a__1::0&comma;a__2::0&comma;b__1::¬0&comma;b__2::¬0&comma;a__1+a__2<cc::0&comma;a__1::0&comma;a__2::¬0&comma;b__1::¬0&comma;b__2::0&comma;a__1+b__2<cc::0&comma;a__1::0&comma;a__2::0&comma;b__1::¬0&comma;b__2::0&comma;a__1+a__2<ca__1+b__2<cc::0&comma;a__1::0&comma;a__2::0&comma;b__1::0&comma;b__2::0&comma;a__1+a__2<ca__1+b__2<cb__1+a__2<cb__1+b__2<cc::0&comma;a__1::0&comma;a__2::¬0&comma;b__1::0&comma;b__2::0&comma;a__1+b__2<cb__1+b__2<cc::0&comma;a__1::0&comma;a__2::0&comma;b__1::0&comma;b__2::¬0&comma;a__1+a__2<cb__1+a__2<c

branch points

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

a__1::¬0&comma;b__1::¬0&comma;z__11&comma;+Ia__2::¬0&comma;b__2::¬0&comma;z__21&comma;+I

branch cuts

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2

a__1::¬0&comma;b__1::¬0&comma;1<z__1a__2::¬0&comma;b__2::¬0&comma;1<z__2

special values

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1

z__1=0z__2=0

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1

a__1=0a__2=0

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1

a__1=0b__2=0

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1

a__2=0b__1=0

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1

b__1=0b__2=0

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a__2,b__2;c;z__2

z__1=0

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a__1,b__1;c;z__1

z__2=0

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a__1,b__1;c;1F23a__2,b__2,ca__1b__1;ca__1,cb__1;z__2

z__1=1

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a__2,b__2;c;1F23a__1,b__1,ca__2b__2;cb__2,ca__2;z__1

z__2=1

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a__2,b__2;c;z__2

a__1=0

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a__1,b__1;c;z__1

a__2=0

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a__2,b__2;c;z__2

b__1=0

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F12a__1,b__1;c;z__1

b__2=0

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__1F121,a__1;a__1+a__2;z__1+z__2F121,a__2;a__1+a__2;z__2z__1z__2+z__1+z__2

b__1=1b__2=1c=a__1+a__2z__1z__2+z__1+z__20

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__1F121,b__1;b__1+a__2;z__1+z__2F121,a__2;b__1+a__2;z__2z__1z__2+z__1+z__2

a__1=1b__2=1c=b__1+a__2z__1z__2+z__1+z__20

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__1F121,a__1;a__1+b__2;z__1+z__2F121,b__2;a__1+b__2;z__2z__1z__2+z__1+z__2

b__1=1a__2=1c=a__1+b__2z__1z__2+z__1+z__20

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1z__1b__2F12a__1+b__2,b__1+b__2;a__1+a__2+b__1+b__2;z__1

c=a__1+a__2+b__1+b__2z__11z__2=z__11+z__1

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1z__1b__2F12a__1+b__2,b__1+b__2;a__1+a__2+b__1+b__2;z__1

c=a__1+a__2+b__1+b__2z__21z__1=z__21+z__2

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1z__1c112z__11a__2F12c2+a__12a__22,c2a__12a__22+12;c;4z__11z__1

z__2=z__12z__112z__11z__11b__1=1a__1b__2=1a__2

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1z__1c112z__11a__2F12c2+a__12a__22,c2a__12a__22+12;c;4z__11z__1

z__1=z__22z__212z__21z__21b__1=1a__1b__2=1a__2

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1z__1c112z__11a__2F12c2+a__12a__22,c2a__12a__22+12;c;4z__11z__1

z__2=z__12z__112z__11z__11a__1=1b__1b__2=1a__2

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=1z__1c112z__11a__2F12c2+a__12a__22,c2a__12a__22+12;c;4z__11z__1

z__2=z__12z__112z__11z__11b__1=1a__1a__2=1b__2

identities

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F3a__1&comma;a__2&comma;b__1&comma;b__2+n&comma;c&comma;z__1&comma;z__2k=1nnka__2kz__2kF3a__1&comma;b__2+k&comma;b__1&comma;k+a__2&comma;k+c&comma;z__1&comma;z__2ck

z__21c::¬0&comma;a__2::0&comma;c<a__2nc

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F3b__1&comma;a__2&comma;a__1+n&comma;b__2&comma;c&comma;z__1&comma;z__2k=1nnkb__1kz__1kF3a__1+k&comma;a__2&comma;k+b__1&comma;b__2&comma;k+c&comma;z__1&comma;z__2ck

z__11c::¬0&comma;b__1::0&comma;c<b__1nc

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=a__1nF3b__1&comma;a__2&comma;a__1+n&comma;b__2&comma;c&comma;z__1&comma;z__2a__1b__1nk=1nnkb__1k−1kF3a__1&comma;a__2&comma;k+b__1&comma;b__2&comma;c&comma;z__1&comma;z__2b__1a__1n+1k

z__11a__1b__1::¬0&comma;na__1b__1b__1a__1n+1::¬0&comma;b__1::0&comma;b__1a__1n+1<b__1nb__1+a__1+n1

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F3b__1&comma;a__2&comma;a__1+n&comma;b__2&comma;c&comma;z__1&comma;z__2b__1z__1k=1nF3a__1+k&comma;a__2&comma;b__1+1&comma;b__2&comma;c+1&comma;z__1&comma;z__2c

z__11c0

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;cn&comma;z__1&comma;z__2a__1b__1z__1k=1nF3a__1+1&comma;a__2&comma;b__1+1&comma;b__2&comma;c+2k&comma;z__1&comma;z__2ckck+1a__2b__2z__2k=1nF3a__1&comma;a__2+1&comma;b__1&comma;b__2+1&comma;c+2k&comma;z__1&comma;z__2ckck+1

z__11z__21c::¬+

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1a__2&comma;b__2&comma;b__1&comma;c&comma;z__2&comma;z__11+z__11z__1b__1

c=a__1+a__2z__11

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1a__2&comma;b__2&comma;a__1&comma;c&comma;z__2&comma;z__11+z__11z__1a__1

c=b__1+a__2z__11

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1b__2&comma;a__2&comma;b__1&comma;c&comma;z__2&comma;z__11+z__11z__1b__1

c=a__1+b__2z__11

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1b__2&comma;a__2&comma;a__1&comma;c&comma;z__2&comma;z__11+z__11z__1a__1

c=b__1+b__2z__11

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1a__1&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__21+z__21z__2b__2

c=a__1+a__2z__21

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1b__1&comma;a__1&comma;b__2&comma;c&comma;z__1&comma;z__21+z__21z__2b__2

c=b__1+a__2z__21

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1a__1&comma;b__1&comma;a__2&comma;c&comma;z__1&comma;z__21+z__21z__2a__2

c=a__1+b__2z__21

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F1b__1&comma;a__1&comma;a__2&comma;c&comma;z__1&comma;z__21+z__21z__2a__2

c=b__1+b__2z__21

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__21+z__1z__1b__1F2b__1+b__2&comma;a__2&comma;b__1&comma;c&comma;b__1+b__2&comma;z__2&comma;1z__21+z__1z__11z__1b__1

c=a__1+a__2z__11

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__21+z__1z__1a__1F2a__1+b__2&comma;a__2&comma;a__1&comma;c&comma;a__1+b__2&comma;z__2&comma;1z__21+z__1z__11z__1a__1

c=b__1+a__2z__11

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__21+z__1z__1b__1F2b__1+a__2&comma;b__2&comma;b__1&comma;c&comma;b__1+a__2&comma;z__2&comma;1z__21+z__1z__11z__1b__1

c=a__1+b__2z__11

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__21+z__1z__1a__1F2a__1+a__2&comma;b__2&comma;a__1&comma;c&comma;a__1+a__2&comma;z__2&comma;1z__21+z__1z__11z__1a__1

c=b__1+b__2z__11

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__11+z__2z__2b__2F2b__1+b__2&comma;a__1&comma;b__2&comma;c&comma;b__1+b__2&comma;z__1&comma;1z__11+z__2z__21z__2b__2

c=a__1+a__2z__21

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__11+z__2z__2b__2F2a__1+b__2&comma;b__1&comma;b__2&comma;c&comma;a__1+b__2&comma;z__1&comma;1z__11+z__2z__21z__2b__2

c=b__1+a__2z__21

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__11+z__2z__2a__2F2b__1+a__2&comma;a__1&comma;a__2&comma;c&comma;b__1+a__2&comma;z__1&comma;1z__11+z__2z__21z__2a__2

c=a__1+b__2z__21

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__11+z__2z__2a__2F2a__1+a__2&comma;b__1&comma;a__2&comma;c&comma;a__1+a__2&comma;z__1&comma;1z__11+z__2z__21z__2a__2

c=b__1+b__2z__21

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F412+b+c2&comma;b&comma;d&comma;e&comma;2z__2gz__2g4gz__2z__2&comma;gg+11+2b+c4g1+z__2z__2g+12b1z__2b

a__2=bb__1=c2b__2=c2e=12+c2a__1=12+c2d=12+b+c2z__1=4gg+122z__2gz__2g4gz__2z__2g+121

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F4b+12c2&comma;b&comma;d&comma;e&comma;2z__2gz__2g4gz__2z__2&comma;gg+12b4g1+z__2z__2g+12b+12c21z__2b+12c2

d=bb__1=c2b__2=c2e=12+c2a__1=12+c2a__2=b+12c2z__1=4gg+122z__2gz__2g4gz__2z__2g+121

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F412+b+c2&comma;b&comma;d&comma;e&comma;f&comma;2z__2fz__2f4fz__2z__2f+11+2b+c4f1+z__2z__2f+12b1z__2b

a__2=bb__1=c2b__2=c2d=12+c2a__1=12+c2e=12+b+c2z__1=4ff+122z__2fz__2f4fz__2z__2f+121

F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=F4b+12c2&comma;b&comma;d&comma;e&comma;f