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AppellF4

The AppellF4 function

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

AppellF4(a,b,c1,c2,z1,z2)

Parameters

a

-

algebraic expression

b

-

algebraic expression

c1

-

algebraic expression

c2

-

algebraic expression

z1

-

algebraic expression

z2

-

algebraic expression

Description

• 

As is the case of all the four multi-parameter Appell functions, AppellF4, 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, AppellF4 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 AppellF4 series and the corresponding domain of convergence can be seen through the FunctionAdvisor

FunctionAdvisordefinition,AppellF4

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=_k1=0_k2=0a_k1+_k2b_k1+_k2z__1_k1z__2_k2c__1_k1c__2_k2_k1!_k2!&comma;z__1+z__2<1

(1)
  

A distinction is made between the AppellF4 doubly hypergeometric series, with the restricted domain of convergence shown above, and the AppellF4 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 AppellF4 variables subscripted with the numbers 1 and 2, the function remains the same; hence

FunctionAdvisorsymmetries&comma;AppellF4

F4a&comma;b&comma;c__2&comma;c__1&comma;z__2&comma;z__1=F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2&comma;F4b&comma;a&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2

(2)
  

Note the existence of another symmetry, also visible in the double sum definition.

  

From the series' definition, AppellF4 is singular (division by zero) when the c1 and/or c2 parameters entering the pochhammer functions in the denominator of the series are non-positive integers, because these pochhammer functions will be equal to zero when the summation index of the series is bigger than the absolute value of the corresponding c1 or c2 parameter.

  

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 AppellF4 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, AppellF4 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 (parameters c1,c2) in the pochhammers 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;AppellF4

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2&comma;c__1::&apos;nonposint&apos;a::¬&apos;nonposint&apos;b::¬&apos;nonposint&apos;c__1::&apos;nonposint&apos;a::&apos;nonposint&apos;b::¬&apos;nonposint&apos;a<c__1c__1::&apos;nonposint&apos;a::¬&apos;nonposint&apos;b::&apos;nonposint&apos;b<c__1c__1::&apos;nonposint&apos;a::&apos;nonposint&apos;b::&apos;nonposint&apos;a<c__1b<c__1c__2::&apos;nonposint&apos;a::¬&apos;nonposint&apos;b::¬&apos;nonposint&apos;c__2::&apos;nonposint&apos;a::&apos;nonposint&apos;b::¬&apos;nonposint&apos;a<c__2c__2::&apos;nonposint&apos;a::¬&apos;nonposint&apos;b::&apos;nonposint&apos;b<c__2c__2::&apos;nonposint&apos;a::&apos;nonposint&apos;b::&apos;nonposint&apos;a<c__2b<c__2

(3)
  

The AppellF4 series is analytically extended to the AppellF4 function defined over the whole complex plane using identities and mainly by integral representations in terms of Eulerian integrals:

FunctionAdvisorintegral_form&comma;AppellF4

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=Γc__101ub1F12a2,12+a2;c__1;4u2z__1z__21+z__1+z__2u21ubc__1+11+z__1z__2ua&DifferentialD;uΓbΓc__1b&comma;z__1+z__2<1c__1=c__20<b0<c__10<c__1+bz__1+z__2<1c__1=c__20<b0<c__20<c__2+b,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=0u2a1F10;c__1;z__1u24F10;c__2;z__2u24&ExponentialE;u&DifferentialD;uΓ2a&comma;z__1+z__2<1b=a+120<az__1+z__2<1z__1z__2<1z__1z__2<1z__1+z__2<1,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=Γc__1Γc__20101ua1vb11uc__1+a+11vc__2+b+11u1+u&alpha;1z__1&comma;z__2+z__2&alpha;1z__1&comma;z__2v&alpha;1z__1&comma;z__2c__1c__2+a+1+b&alpha;1z__1&comma;z__2z__2&alpha;1z__1&comma;z__2c__1c__2+a+11v&alpha;1z__1&comma;z__2c__1c__2+b+1&DifferentialD;u&DifferentialD;vΓaΓbΓc__1aΓc__2b&comma;z__1+z__2<10<a0<b0<c__1+a0<c__2+bαz__1&comma;z__22+z__1z__21αz__1&comma;z__2+z__2=0,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=Γc__2Γc__10101ua1vb11uc__2+a+11vbc__1+11u1+u&alpha;1z__2&comma;z__1+z__1&alpha;1z__2&comma;z__1v&alpha;1z__2&comma;z__1c__1c__2+a+1+b&alpha;1z__2&comma;z__1z__1&alpha;1z__2&comma;z__1c__1c__2+a+11v&alpha;1z__2&comma;z__1c__1c__2+b+1&DifferentialD;u&DifferentialD;vΓaΓbΓc__2aΓc__1b&comma;z__1+z__2<10<a0<b0<c__2+a0<c__1+bαz__1&comma;z__22+z__1z__21αz__1&comma;z__2+z__2=0

(4)
  

AppellF4 is the only one among the four Appell functions that has no single integral representation in the general case (all of its parameters arbitrary). These integral representations are also the starting point for the derivation of many of the identities known for AppellF4.

  

AppellF4 also satisfies a linear system of partial differential equations of second order

FunctionAdvisorDE&comma;AppellF4

fa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2&comma;2z__12fa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=2z__22z__1z__2fa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2z__11z__222z__22fa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2z__1z__11+ab1z__1+c__1z__1fa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2z__1z__11z__2a+b+1z__2fa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2z__1z__11abfa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2z__1z__11&comma;2z__1z__2fa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=z__12z__12fa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__22z__2z__212z__22fa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__22z__1a+b+1z__1fa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__22z__2+ab1z__2+c__2z__2fa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__22z__1z__2abfa&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__22z__1z__2

(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 AppellF4. For example, the fourteen polynomial cases of AppellF4 are

AppellF4:-SpecialValues:-Polynomial

8,a&comma;b&comma;c1&comma;c2&comma;z1&comma;z2a::&apos;nonposint&apos;&comma;c1::¬&apos;nonposint&apos;&comma;c2::¬&apos;nonposint&apos;&comma;b::&apos;nonposint&apos;&comma;c1::¬&apos;nonposint&apos;&comma;c2::¬&apos;nonposint&apos;&comma;a::&apos;nonposint&apos;&comma;c1::&apos;nonposint&apos;&comma;c2::¬&apos;nonposint&apos;&comma;c1a&comma;a::&apos;nonposint&apos;&comma;c1::¬&apos;nonposint&apos;&comma;c2::&apos;nonposint&apos;&comma;c2a&comma;a::&apos;nonposint&apos;&comma;c1::&apos;nonposint&apos;&comma;c2::&apos;nonposint&apos;&comma;c1a&comma;c2a&comma;b::&apos;nonposint&apos;&comma;c1::&apos;nonposint&apos;&comma;c2::¬&apos;nonposint&apos;&comma;c1b&comma;b::&apos;nonposint&apos;&comma;c1::¬&apos;nonposint&apos;&comma;c2::&apos;nonposint&apos;&comma;c2b&comma;b::&apos;nonposint&apos;&comma;c1::&apos;nonposint&apos;&comma;c2::&apos;nonposint&apos;&comma;c1b&comma;c2b

(6)

Likewise, the conditions for the singular cases of AppellF4 can be seen either using the FunctionAdvisor or entering AppellF4:-Singularities(), so with no arguments.

For particular values of its parameters, AppellF4 is related to the hypergeometric function. These hypergeometric cases are returned automatically. For example, for c1=c2,z1=z2,

%AppellF4=AppellF4a&comma;b&comma;c__2&comma;c__2&comma;z__2&comma;z__2

%AppellF4a&comma;b&comma;c__2&comma;c__2&comma;z__2&comma;z__2=F34a2,b2,a2+12,b2+12;c__2,c__22,c__22+12;4z__22

(7)

To see all the hypergeometric cases, enter

FunctionAdvisorspecialize&comma;AppellF4&comma;hypergeom

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F12a,b;c__2;z__2&comma;z__1=0,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F12a,b;c__1;z__1&comma;z__2=0,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F12a,a+12;c__1;z__11+z__2221+z__22a+F12a,a+12;c__1;z__1z__21221z__22a&comma;b=a+12c__2=12,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F12a,a+12;c__2;z__21+z__1221+z__12a+F12a,a+12;c__2;z__2z__11221z__12a&comma;b=a+12c__1=12,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F12b,b+12;c__1;z__11+z__2221+z__22b+F12b,b+12;c__1;z__1z__21221z__22b&comma;a=b+12c__2=12,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F12b,b+12;c__2;z__21+z__1221+z__12b+F12b,b+12;c__2;z__2z__11221z__12b&comma;a=b+12c__1=12,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=z__2+1z__14z__1z__2+1+z__1+z__222z__2az__1+1z__24z__1z__2+1+z__1+z__222z__1aF12a,ab+1;b;1+z__1+z__2+4z__1z__2+1+z__1+z__2224z__1z__2&comma;c__1=bc__2=b,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=z__2+1z__14z__1z__2+1+z__1+z__222z__2bz__1+1z__24z__1z__2+1+z__1+z__222z__1bF12b,ba+1;a;1+z__1+z__2+4z__1z__2+1+z__1+z__2224z__1z__2&comma;c__1=ac__2=a,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F12a,b;ab+1;1+z__1+z__2+4z__1z__2+1+z__1+z__22z__1+1z__24z__1z__2+1+z__1+z__222z__1z__21+z__1+4z__1z__2+1+z__1+z__22z__1+1z__24z__1z__2+1+z__1+z__222z__1a&comma;c__1=ab+1c__2=b,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F12a,b;ab+1;1+z__1+z__2+4z__1z__2+1+z__1+z__22z__21+z__1+4z__1z__2+1+z__1+z__222z__2z__1+1z__24z__1z__2+1+z__1+z__22z__2+1z__14z__1z__2+1+z__1+z__222z__2a&comma;c__2=ab+1c__1=b,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F12a,b;ba+1;1+z__1+z__2+4z__1z__2+1+z__1+z__22z__1+1z__24z__1z__2+1+z__1+z__222z__1z__21+z__1+4z__1z__2+1+z__1+z__22z__1+1z__24z__1z__2+1+z__1+z__222z__1b&comma;c__1=ba+1c__2=a,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F12a,b;ba+1;1+z__1+z__2+4z__1z__2+1+z__1+z__22z__21+z__1+4z__1z__2+1+z__1+z__222z__2z__1+1z__24z__1z__2+1+z__1+z__22z__2+1z__14z__1z__2+1+z__1+z__222z__2b&comma;c__2=ba+1c__1=a,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F34a,b,c__12+c__22,c__12+c__2212;c__1,c__2,c__1+c__21;4z__1&comma;z__1=z__2,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F34a2,b2,a2+12,b2+12;c__1,c__12,c__12+12;4z__12&comma;z__2=z__1c__1=c__2

(8)

Other special values of AppellF4 can be seen using FunctionAdvisor(special_values, AppellF4).

By requesting the sum form of AppellF4, 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;AppellF4

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=m=0n=0am+nbm+nz__1mz__2nc__1mc__2nm!n!&comma;z__2+z__1<1,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=k=0akbkF12a+k,b+k;c__2;z__2z__1kc__1kk!&comma;z__2+z__1<1,F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=k=0akbkF12a+k,b+k;c__1;z__1z__2kc__2kk!&comma;z__2+z__1<1

(9)

As indicated in the formulas above, for AppellF4 (also for AppellF2), and unlike the case of AppellF1 and AppellF3, the domain of convergence with regards to the two variables z1 and z2 is entangled, i.e. it intrinsically depends on a combination of the two variables, so the hypergeometric coefficient in one variable in the single sum form does not extend the domain of convergence of the double sum but for particular cases, and from the formulas above one cannot conclude about the value of the function when one of z1 or z2 is equal to 1 unless the other one is exactly equal to 0.

AppellF4 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,

AppellF4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2 &equals; AppellF4:-TransformationsEuler1a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=Γc__2Γbaz__2aF4a&comma;ac__2+1&comma;ab+1&comma;c__1&comma;1z__2&comma;z__1z__2Γc__2aΓb+Γc__2Γabz__2bF4b&comma;bc__2+1&comma;ba+1&comma;c__1&comma;1z__2&comma;z__1z__2Γc__2bΓa

(10)

Among other situations, this identity is useful when the sum of the square roots of the absolute values of z1 and z2 is larger than 1 but the same sum constructed with the arguments in the same position of AppellF4 on the right-hand side is smaller than 1. Another case where this identity is useful is when z1=1, so that the two AppellF4 functions on the right-hand side will have the two main variables (last arguments) equal, in turn a special value of hypergeometric 4F3 type:

eval&comma; z__1 &equals; 1

F4a&comma;b&comma;c__1&comma;c__2&comma;1&comma;z__2=Γc__2Γbaz__2aF34a,ac__2+1,a2b2+c__12,a2b2+12+c__12;c__1,ab+1,ab+c__1;4z__2Γc__2aΓb+Γc__2Γabz__2bF34b,bc__2+1,b2a2+c__12,b2a2+12+c__12;c__1,ba+1,ba+c__1;4z__2Γc__2bΓa

(11)

This formula analytically extends to the whole complex plane the AppellF4 series when any of z1=1 or z2=1 (the latter using the symmetry of AppellF4 - see the beginning of the Description section).

A contiguity transformation for AppellF4

AppellF4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2 &equals; AppellF4:-TransformationsContiguity1a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F4b&comma;a+1&comma;c__1&comma;c__2&comma;z__1&comma;z__2aabbF4a&comma;b+1&comma;c__1&comma;c__2&comma;z__1&comma;z__2ab

(12)

The contiguity transformations available in this way are

indicesAppellF4:-TransformationsContiguity

1,2,3,4

(13)

By using differential algebra techniques, the PDE system satisfied by AppellF4 can be transformed into an equivalent PDE system where one of the equations is a fourth order linear ODE in z2 parametrized by z1. This linear ODE has four regular singularities, some of which depend on z1 and the function's parameters. These singularities can be see directly from the function using the MathematicalFunctions:-Evalf:-Singularities command

MathematicalFunctions:-Evalf:-SingularitiesAppellF4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2

0&comma;z__11a+bc__1+1a+bc__12c__2+3c__11b+ac__1+1b+a&comma;z__1+12z__1&comma;z__1+1+2z__1&comma;+I

(14)

You can also see a general presentation of AppellF4, organized into sections and including plots, using the FunctionAdvisor

FunctionAdvisorAppellF4

AppellF4

describe

AppellF4=Appell 2-variable hypergeometric function F4

definition

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=_k1=0_k2=0a_k1+_k2b_k1+_k2z__1_k1z__2_k2c__1_k1c__2_k2_k1!_k2!

z__1+z__2<1

classify function

Appell

symmetries

F4a&comma;b&comma;c__2&comma;c__1&comma;z__2&comma;z__1=F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2

F4b&comma;a&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2

plot

singularities

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2

c__1::Typesetting:-_Hold&apos;nonposint&apos;a::¬Typesetting:-_Hold&apos;nonposint&apos;b::¬Typesetting:-_Hold&apos;nonposint&apos;c__1::Typesetting:-_Hold&apos;nonposint&apos;a::Typesetting:-_Hold&apos;nonposint&apos;b::¬Typesetting:-_Hold&apos;nonposint&apos;a<c__1c__1::Typesetting:-_Hold&apos;nonposint&apos;a::¬Typesetting:-_Hold&apos;nonposint&apos;b::Typesetting:-_Hold&apos;nonposint&apos;b<c__1c__1::Typesetting:-_Hold&apos;nonposint&apos;a::Typesetting:-_Hold&apos;nonposint&apos;b::Typesetting:-_Hold&apos;nonposint&apos;a<c__1b<c__1c__2::Typesetting:-_Hold&apos;nonposint&apos;a::¬Typesetting:-_Hold&apos;nonposint&apos;b::¬Typesetting:-_Hold&apos;nonposint&apos;c__2::Typesetting:-_Hold&apos;nonposint&apos;a::Typesetting:-_Hold&apos;nonposint&apos;b::¬Typesetting:-_Hold&apos;nonposint&apos;a<c__2c__2::Typesetting:-_Hold&apos;nonposint&apos;a::¬Typesetting:-_Hold&apos;nonposint&apos;b::Typesetting:-_Hold&apos;nonposint&apos;b<c__2c__2::Typesetting:-_Hold&apos;nonposint&apos;a::Typesetting:-_Hold&apos;nonposint&apos;b::Typesetting:-_Hold&apos;nonposint&apos;a<c__2b<c__2

branch points

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2

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

branch cuts

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2

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

special values

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=1

z__1=0z__2=0

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=1

a=0

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=1

b=0

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F12a,b;c__2;z__2

z__1=0

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F12a,b;c__1;z__1

z__2=0

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=1+z__22aF12a,a+12;c__1;z__11+z__222+1z__22aF12a,a+12;c__1;z__11z__222

b=a+12c__2=12

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=1+z__12aF12a,a+12;c__2;z__21+z__122+1z__12aF12a,a+12;c__2;z__21z__122

b=a+12c__1=12

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=1+z__22bF12b,b+12;c__1;z__11+z__222+1z__22bF12b,b+12;c__1;z__11z__222

a=b+12c__2=12

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=1+z__12bF12b,b+12;c__2;z__21+z__122+1z__12bF12b,b+12;c__2;z__21z__122

a=b+12c__1=12

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=11+z__1+z__2+4z__1z__2+1+z__1+z__222z__2a11+z__1+z__2+4z__1z__2+1+z__1+z__222z__1aF12a,ab+1;b;1+z__1+z__2+4z__1z__2+1+z__1+z__2224z__1z__2

c__1=bc__2=b

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=11+z__1+z__2+4z__1z__2+1+z__1+z__222z__2b11+z__1+z__2+4z__1z__2+1+z__1+z__222z__1bF12b,ba+1;a;1+z__1+z__2+4z__1z__2+1+z__1+z__2224z__1z__2

c__1=ac__2=a

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=11+z__1+z__2+4z__1z__2+1+z__1+z__222z__1aF12a,b;ab+1;1+z__1+z__2+4z__1z__2+1+z__1+z__2211+z__1+z__2+4z__1z__2+1+z__1+z__222z__12z__211+z__1+z__2+4z__1z__2+1+z__1+z__222z__2

c__1=ab+1c__2=b

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=11+z__1+z__2+4z__1z__2+1+z__1+z__222z__2aF12a,b;ab+1;1+z__1+z__2+4z__1z__2+1+z__1+z__2211+z__1+z__2+4z__1z__2+1+z__1+z__222z__22z__111+z__1+z__2+4z__1z__2+1+z__1+z__222z__1

c__2=ab+1c__1=b

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=11+z__1+z__2+4z__1z__2+1+z__1+z__222z__1bF12a,b;ba+1;1+z__1+z__2+4z__1z__2+1+z__1+z__2211+z__1+z__2+4z__1z__2+1+z__1+z__222z__12z__211+z__1+z__2+4z__1z__2+1+z__1+z__222z__2

c__1=ba+1c__2=a

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=11+z__1+z__2+4z__1z__2+1+z__1+z__222z__2bF12a,b;ba+1;1+z__1+z__2+4z__1z__2+1+z__1+z__2211+z__1+z__2+4z__1z__2+1+z__1+z__222z__22z__111+z__1+z__2+4z__1z__2+1+z__1+z__222z__1

c__2=ba+1c__1=a

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F34a,b,c__12+c__22,c__12+c__2212;c__1,c__2,c__1+c__21;4z__1

z__1=z__2

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F34a2,b2,a2+12,b2+12;c__1,c__12,c__12+12;4z__12

z__2=z__1c__1=c__2

identities

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F4b&comma;a+1&comma;c__1&comma;c__2&comma;z__1&comma;z__2ab+abF4a&comma;b+1&comma;c__1&comma;c__2&comma;z__1&comma;z__2b+a

abz__11z__21

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F4b&comma;a+n&comma;c__1&comma;c__2&comma;z__1&comma;z__2bz__1k=1nF4a+k&comma;b+1&comma;c__1+1&comma;c__2&comma;z__1&comma;z__2c__1bz__2k=1nF4a+k&comma;b+1&comma;c__1&comma;c__2+1&comma;z__1&comma;z__2c__2

z__11z__21c__10c__20

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F4b&comma;a+n&comma;c__1&comma;c__2&comma;z__1&comma;z__2i=0nk=0niF4a+i+k&comma;k+i+b&comma;i+c__1&comma;k+c__2&comma;z__1&comma;z__2ninikbk+iz__1iz__2kc__1ic__2k+F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2

z__11z__21c__1::¬Typesetting:-_Hold&apos;nonposint&apos;nc__1c__1::¬Typesetting:-_Hold&apos;nonposint&apos;nc__2

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F4a&comma;b&comma;c__1n&comma;c__2&comma;z__1&comma;z__2abz__1k=0n1F4a+1&comma;b+1&comma;c__1+1k&comma;c__2&comma;z__1&comma;z__2c__1kc__1k1

z__11z__21c__1::¬Typesetting:-_Hold&apos;nonnegint&apos;n1<c__1

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=Γc__2Γbaz__2aF4a&comma;ac__2+1&comma;ab+1&comma;c__1&comma;1z__2&comma;z__1z__2Γc__2aΓb+Γc__2Γb+az__2bF4b&comma;bc__2+1&comma;ba+1&comma;c__1&comma;1z__2&comma;z__1z__2Γc__2bΓa

z__20a::¬Typesetting:-_Hold&apos;nonposint&apos;b::¬Typesetting:-_Hold&apos;nonposint&apos;c__2::¬Typesetting:-_Hold&apos;nonposint&apos;c__2a::¬Typesetting:-_Hold&apos;nonposint&apos;c__2b::¬Typesetting:-_Hold&apos;nonposint&apos;ba::¬Typesetting:-_Hold&apos;integer&apos;

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F112+ab&comma;b+a&comma;b&comma;2a+12b&comma;4z__2z__2+12&comma;4z__2z__2+z__12z__21z__2+12a1z__1z__2+12b

c__1=ac__2=ab+1

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F112+ba&comma;ba&comma;a&comma;2b+12a&comma;4z__2z__2+12&comma;4z__2z__2+z__12z__21z__2+12b1z__1z__2+12a

c__1=bc__2=ba+1

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F112+ab&comma;b+a&comma;b&comma;2a+12b&comma;4z__1z__1+12&comma;4z__1z__1+z__22z__11z__1+12a1z__2z__1+12b

c__2=ac__1=ab+1

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F112+ba&comma;ba&comma;a&comma;2b+12a&comma;4z__1z__1+12&comma;4z__1z__1+z__22z__11z__1+12b1z__2z__1+12a

c__2=bc__1=ba+1

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F2a&comma;b&comma;12+ab&comma;c__1&comma;2a+12b&comma;z__1z__2+12&comma;4z__2z__2+12z__2+12a

c__2=ab+1

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F2b&comma;a&comma;12+ba&comma;c__1&comma;2b+12a&comma;z__1z__2+12&comma;4z__2z__2+12z__2+12b

c__2=ba+1

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F2a&comma;b&comma;12+ab&comma;c__2&comma;2a+12b&comma;z__2z__1+12&comma;4z__1z__1+12z__1+12a

c__1=ab+1

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=F2b&comma;a&comma;12+ba&comma;c__2&comma;2b+12a&comma;z__2z__1+12&comma;4z__1z__1+12z__1+12b

c__1=ba+1

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=Physics:−Library:−Addak1+k2F2b&comma;k1&comma;k2&comma;c__1&comma;c__2&comma;z__1&comma;z__2k1!k2!&comma;k1+k2a−1a

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

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=z__2+2z__2z__1+1z__22z__2z__1+1bF3b+a&comma;b&comma;12+ab&comma;12+ab&comma;2a+12b&comma;4z__2z__2+12&comma;4z__2z__2+2z__2+z__11z__2+12a1z__1z__2+12b

c__2=ab+1c__1=az__1z__2+121

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=z__2+2z__2z__1+1z__22z__2z__1+1aF3ba&comma;a&comma;12+ba&comma;12+ba&comma;2b+12a&comma;4z__2z__2+12&comma;4z__2z__2+2z__2+z__11z__2+12b1z__1z__2+12a

c__2=ba+1c__1=bz__1z__2+121

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=z__1+2z__1z__2+1z__12z__1z__2+1bF3b+a&comma;b&comma;12+ab&comma;12&pl