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

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : System : Information : Updates : Maple 2017 : Mathematical Functions

Mathematical Functions

Relevant developments in the MathematicalFunctions project happened for Maple 2017, regarding both the addition of the four Appell functions, representing the first ever full implementation of these functions in computational environments, as well as the addition of a new package, Evalf, for performing numerical experimentation taking advantage of sophisticated symbolic computation functionality. The Evalf package and project aims to provide a user-friendly environment to develop and work with numerical algorithms for mathematical functions.

 

The Four Appell Functions

The Evalf Package

The Four Appell Functions

The four multi-parameter Appell functions, AppellF1, AppellF2, AppellF3 and AppellF4 are doubly hypergeometric functions that include as particular cases the 2F1 hypergeometric and some cases of the MeijerG function, and with them most of the known functions of mathematical physics. These Appell functions have been popping up with increasing frequency in applications in quantum mechanics, molecular physics, and general relativity.

As in the case of the hypergeometric function, a distinction is made between the four Appell series, with restricted domain of convergence, and the four Appell functions, that coincide with the series in their domain of convergence but also extend them analytically to the whole complex plane. The Maple implementation of the Appell functions includes a thorough set of their symbolic properties, all accessible using the FunctionAdvisor, as well as numerical algorithms to evaluate the four functions over the whole complex plane, representing the first ever complete computational implementation of these functions.

To display special functions and sequences using textbook notation as shown in this page, use extended typesetting and enable the typesetting of mathematical functions

interfacetypesetting = extended: Typesetting:-EnableTypesetRuleTypesetting:-SpecialFunctionRules:

Examples

  

The definition of the four Appell series and the corresponding domains of convergence can be seen through the FunctionAdvisor. For example,

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

(1.1.1)

FunctionAdvisordefinition&comma; AppellF2&semi;

F2a&comma;b__1&comma;b__2&comma;c__1&comma;c__2&comma;z__1&comma;z__2=_k1=0_k2=0a_k1+_k2b__1_k1b__2_k2z__1_k1z__2_k2c__1_k1c__2_k2_k1!_k2!&comma;z__1+z__2<1

(1.1.2)

FunctionAdvisordefinition&comma; AppellF3&semi;

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

(1.1.3)

FunctionAdvisordefinition&comma; AppellF4&semi;

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

(1.1.4)
  

From these definitions, these series and the corresponding analytic extensions (Appell functions) are singular (division by zero) when the c parameters entering the pochhammer functions in the denominators of these series are non-positive integers. For an analogous reason, when the a and/or b parameters entering the pochhammer functions in the numerators of the series are non-positive integers, the series will truncate and the Appell functions will be polynomial. Consult the FunctionAdvisor for comprehensive information on the combinations of all these conditions. For example, for AppellF1, 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

(1.1.5)
  

 

  

By requesting the sum form of the Appell functions, besides their double power series definition, we also see the particular form the four series take when one of the summations is performed and the result expressed in terms of 2F1 hypergeometric functions. For example, for AppellF3,

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

(1.1.6)
  

So, for AppellF3 (and also for AppellF1, but not for AppellF2 nor AppellF4) 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.

  

In the literature, the Appell series are analytically extended by integral representations in terms of Eulerian double integrals. With the exception of AppellF4, one of the two iterated integrals can always be computed resulting in a single integral with hypergeometric integrand. For example, for AppellF2

FunctionAdvisorintegral_form&comma; AppellF2&semi;

F2a&comma;b__1&comma;b__2&comma;c__1&comma;c__2&comma;z__1&comma;z__2=Γc__101ub__11F12a,b__2;c__2;z__2uz__111uc__1+b__1+1uz__1+1a&DifferentialD;uΓb__1Γc__1b__1&comma;z__110<b__10<c__1+b__1,F2a&comma;b__1&comma;b__2&comma;c__1&comma;c__2&comma;z__1&comma;z__2=Γc__201ub__21F12a,b__1;c__1;z__1uz__211u1+b__2c__2uz__2+1a&DifferentialD;uΓb__2Γc__2b__2&comma;z__210<b__20<c__2+b__2,F2a&comma;b__1&comma;b__2&comma;c__1&comma;c__2&comma;z__1&comma;z__2=Γc__1Γc__20101ub__11vb__211uc__1+b__1+11v1+b__2c__2uz__1vz__2+1a&DifferentialD;u&DifferentialD;vΓb__1Γb__2Γc__1b__1Γc__2b__2&comma;0<b__10<b__20<c__1+b__10<c__2+b__2,F2a&comma;b__1&comma;b__2&comma;c__1&comma;c__2&comma;z__1&comma;z__2=0ua1F11b__1;c__1;uz__1F11b__2;c__2;uz__2&ExponentialE;u&DifferentialD;uΓa&comma;z__1+z__2<10<a

(1.1.7)
  

For the purpose of numerically evaluating the four Appell functions over the whole complex plane, instead of numerically evaluating the integral representations, it is simpler, when possible, to evaluate the function using identities. For example, with the exception of AppellF3, the Appell functions admit identities analogous to Euler identities for the hypergeometric function. These Euler-type identities, as well as contiguity identities for the four Appell functions, are visible using the FunctionAdvisor with the option identities, or directly from the function. For AppellF4, for instance, provided that none of a, b, ab, c2a is a non-positive integer,

F4a&comma;b&comma;c__1&comma;c__2&comma;z__1&comma;z__2=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;1+bc__2&comma;ba+1&comma;c__1&comma;1z__2&comma;z__1z__2Γc__2bΓa

(1.1.8)
  

and this identity can be used to evaluate AppellF4 at z1&equals;1 over the whole complex plane since, in that case, the two variables of the Appell Functions on right-hand side become equal, and that is a special case of AppellF4, expressible in terms of hypergeometric 4F3 functions

z__1=1|z__1=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,1+bc__2,b2a2+c__12,b2a2+12+c__12;c__1,ba+1,ba+c__1;4z__2Γc__2bΓa

(1.1.9)

A plot of the AppellF2 function for some values of its parameters

F2  AppellF2expIz&comma;12I&comma;37I4&comma;4&comma;57&plus;6I&comma;6&comma;z

F2F2&ExponentialE;Iz&comma;I2&comma;37I4&comma;4&comma;57+6I&comma;6&comma;z

(1.1.10)

plotRe&comma;ImF2&comma;z&equals;1..1

  

A thorough set with the main symbolic properties of any of the four Appell functions, for instance for AppellF3, can be seen via

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::Typesetting:-_Hold&apos;nonposint&apos;a__1::¬Typesetting:-_Hold&apos;nonposint&apos;b__1::¬Typesetting:-_Hold&apos;nonposint&apos;c::Typesetting:-_Hold&apos;nonposint&apos;a__2::¬Typesetting:-_Hold&apos;nonposint&apos;b__2::¬Typesetting:-_Hold&apos;nonposint&apos;c::Typesetting:-_Hold&apos;nonposint&apos;a__1::¬Typesetting:-_Hold&apos;nonposint&apos;a__2::Typesetting:-_Hold&apos;nonposint&apos;b__1::Typesetting:-_Hold&apos;nonposint&apos;b__2::¬Typesetting:-_Hold&apos;nonposint&apos;a__2+b__1<cc::Typesetting:-_Hold&apos;nonposint&apos;a__1::¬Typesetting:-_Hold&apos;nonposint&apos;a__2::¬Typesetting:-_Hold&apos;nonposint&apos;b__1::Typesetting:-_Hold&apos;nonposint&apos;b__2::Typesetting:-_Hold&apos;nonposint&apos;b__1+b__2<cc::Typesetting:-_Hold&apos;nonposint&apos;a__1::¬Typesetting:-_Hold&apos;nonposint&apos;a__2::Typesetting:-_Hold&apos;nonposint&apos;b__1::Typesetting:-_Hold&apos;nonposint&apos;b__2::Typesetting:-_Hold&apos;nonposint&apos;a__2+b__1<cb__1+b__2<cc::Typesetting:-_Hold&apos;nonposint&apos;a__1::Typesetting:-_Hold&apos;nonposint&apos;a__2::Typesetting:-_Hold&apos;nonposint&apos;b__1::¬Typesetting:-_Hold&apos;nonposint&apos;b__2::¬Typesetting:-_Hold&apos;nonposint&apos;a__1+a__2<cc::Typesetting:-_Hold&apos;nonposint&apos;a__1::Typesetting:-_Hold&apos;nonposint&apos;a__2::¬Typesetting:-_Hold&apos;nonposint&apos;b__1::¬Typesetting:-_Hold&apos;nonposint&apos;b__2::Typesetting:-_Hold&apos;nonposint&apos;a__1+b__2<cc::Typesetting:-_Hold&apos;nonposint&apos;a__1::Typesetting:-_Hold&apos;nonposint&apos;a__2::Typesetting:-_Hold&apos;nonposint&apos;b__1::¬Typesetting:-_Hold&apos;nonposint&apos;b__2::Typesetting:-_Hold&apos;nonposint&apos;a__1+a__2<ca__1+b__2<cc::Typesetting:-_Hold&apos;nonposint&apos;a__1::Typesetting:-_Hold&apos;nonposint&apos;a__2::Typesetting:-_Hold&apos;nonposint&apos;b__1::Typesetting:-_Hold&apos;nonposint&apos;b__2::Typesetting:-_Hold&apos;nonposint&apos;a__1+a__2<ca__1+b__2<ca__2+b__1<cb__1+b__2<cc::Typesetting:-_Hold&apos;nonposint&apos;a__1::Typesetting:-_Hold&apos;nonposint&apos;a__2::¬Typesetting:-_Hold&apos;nonposint&apos;b__1::Typesetting:-_Hold&apos;nonposint&apos;b__2::Typesetting:-_Hold&apos;nonposint&apos;a__1+b__2<cb__1+b__2<cc::Typesetting:-_Hold&apos;nonposint&apos;a__1::Typesetting:-_Hold&apos;nonposint&apos;a__2::Typesetting:-_Hold&apos;nonposint&apos;b__1::Typesetting:-_Hold&apos;nonposint&apos;b__2::¬Typesetting:-_Hold&apos;nonposint&apos;a__1+a__2<ca__2+b__1<c

branch points

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

a__1::¬Typesetting:-_Hold&apos;nonposint&apos;b__1::¬Typesetting:-_Hold&apos;nonposint&apos;z__11&comma;+Ia__2::¬Typesetting:-_Hold&apos;nonposint&apos;b__2::¬Typesetting:-_Hold&apos;nonposint&apos;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::¬Typesetting:-_Hold&apos;nonposint&apos;b__1::¬Typesetting:-_Hold&apos;nonposint&apos;1<z__1a__2::¬Typesetting:-_Hold&apos;nonposint&apos;b__2::¬Typesetting:-_Hold&apos;nonposint&apos;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;cb__1,ca__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;ca__2,cb__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;a__2+b__1;z__1+z__2F121,a__2;a__2+b__1;z__2z__1z__2+z__1+z__2

a__1=1b__2=1c=a__2+b__1z__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::¬Typesetting:-_Hold&apos;nonposint&apos;a__2::Typesetting:-_Hold&apos;nonposint&apos;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::¬Typesetting:-_Hold&apos;nonposint&apos;b__1::Typesetting:-_Hold&apos;nonposint&apos;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::¬Typesetting:-_Hold&apos;nonposint&apos;na__1b__1b__1a__1n+1::¬Typesetting:-_Hold&apos;nonposint&apos;b__1::Typesetting:-_Hold&apos;nonposint&apos;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::¬Typesetting:-_Hold&apos;posint&apos;

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__2z__21z__2+1b__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__2z__21z__2+1b__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__2z__21z__2+1a__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__2z__21z__2+1a__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__1z__21z__2b__2F2b__1+b__2&comma;a__1&comma;b__2&comma;c&comma;b__1+b__2&comma;z__1&comma;1z__1z__21z__2z__2+1b__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__1z__21z__2b__2F2a__1+b__2&comma;b__1&comma;b__2&comma;c&comma;a__1+b__2&comma;z__1&comma;1z__1z__21z__2z__2+1b__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__1z__21z__2a__2F2b__1+a__2&comma;a__1&comma;a__2&comma;c&comma;b__1+a__2&comma;z__1&comma;1z__1z__21z__2z__2+1a__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__1z__21z__2a__2F2a__1+a__2&comma;b__1&comma;a__2&comma;c&comma;a__1+a__2&comma;z__1&comma;1z__1z__21z__2z__2+1a__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+c4gz__21z__2g+12bz__2+1b

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+12b4gz__21z__2g+12b+12c2z__2+1b+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+c4fz__21z__2f+12bz__2+1b

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&comma;2z__2fz__2f4fz__2z__2f+12b4fz__21z__2f+12b+12c2z__2+1b+12c2

e=bb__1=c2b__2=c2d=12+c2a__1=12+c2a__2=b+12c2z__1=4ff+122z__2fz__2f4fz__2z__2f+121

sum form

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!

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!

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!

z__2<1

series

seriesF3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2&comma;z__1&comma;4=F12a__2,b__2;c;z__2+a__1b__1F12a__2,b__2;c+1;z__2cz__1+12a__1b__1a__1+1b__1+1F12a__2,b__2;c+2;z__2cc+1z__12+16a__1b__1a__1+1b__1+1a__1+2b__1+2F12a__2,b__2;c+3;z__2cc+1c+2z__13+Oz__14

seriesF3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2&comma;z__2&comma;4=F12a__1,b__1;c;z__1+a__2b__2F12a__1,b__1;c+1;z__1cz__2+12a__2b__2a__2+1b__2+1F12a__1,b__1;c+2;z__1cc+1z__22+16a__2b__2a__2+1b__2+1a__2+2b__2+2F12a__1,b__1;c+3;z__1cc+1c+2z__23+Oz__24

integral form

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+z__1u1a__1&DifferentialD;uΓb__1Γcb__1

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;uz__1uc+b__2+11+u1z__2a__2&DifferentialD;uΓb__2Γcb__2

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;uz__1F12a__2,b__2;cρ;u1z__2&DifferentialD;uΓρΓcρ

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+1uz__1+1a__1vz__2+1a__2&DifferentialD;u&DifferentialD;vΓb__1Γb__2Γcb__1b__2

0<b__10<b__20<c+b__1+b__2

differentiation rule

z__1F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=a__1b__1F3a__1+1&comma;a__2&comma;b__1+1&comma;b__2&comma;c+1&comma;z__1&comma;z__2c

nz__1nF3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=a__1nb__1nF3n+a__1&comma;a__2&comma;n+b__1&comma;b__2&comma;n+c&comma;z__1&comma;z__2cn

z__2F3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=a__2b__2F3a__1&comma;a__2+1&comma;b__1&comma;b__2+1&comma;c+1&comma;z__1&comma;z__2c

nz__2nF3a__1&comma;a__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=a__2nb__2nF3a__1&comma;n+a__2&comma;b__1&comma;n+b__2&comma;n+c&comma;z__1&comma;z__2cn

DE

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

2z__12fa__2&comma;b__1&comma;b__2&comma;c&comma;z__1&comma;z__2=z__22z__2z__1fa__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