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The four Appell functions

The four multi-parameter Appell functions, AppellF1, AppellF2, AppellF3, and AppellF4

 

Description

Definition and Symmetries

Polynomial and Singular Cases

Single Power Series with Hypergeometric Coefficients

Analytic Extension from the Appell Series to the Appell Functions

Euler-Type and Contiguity Identities

Appell Differential Equations

References

Compatibility

Description

  

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.

  

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

Typesetting:-EnableTypesetRuleTypesetting:-SpecialFunctionRules:

Definition and Symmetries

  

The definition of the four Appell series and the corresponding domains of convergence can be seen through the FunctionAdvisor. For example, for AppellF1,

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)
  

From this definition, 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)
  

Analogously, the definition and symmetry of AppellF2 are

FunctionAdvisordefinition&comma;AppellF2

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

(3)

FunctionAdvisorsymmetries&comma;AppellF2

F2a&comma;b__2&comma;b__1&comma;c__2&comma;c__1&comma;z__2&comma;z__1=F2a&comma;b__1&comma;b__2&comma;c__1&comma;c__2&comma;z__1&comma;z__2

(4)
  

The cases of AppellF3 and AppellF4 are more general in that, from their definition, besides the symmetry under a swap of the subscripted variables, these two functions have additional symmetries under exchange of positions of the function's parameters

FunctionAdvisordefinition&comma;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

(5)

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

(6)

FunctionAdvisordefinition&comma;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

(7)

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

(8)

Polynomial and Singular Cases

  

From these four definitions, the 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: these pochhammer functions will be equal to zero when the summation indices of these series are bigger than the absolute values of the c parameters.

  

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. As is the case of the hypergeometric function, when both the pochhammers in the numerators and denominators have non-positive integer arguments, the Appell functions are polynomial when the absolute values of the non-positive integers in the numerators are smaller than or equal to the absolute values of the non-positive integers in the denominators, and singular otherwise. The combinatorial of all these conditions can also be consulted using the FunctionAdvisor. 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

(9)
  

The conditions for both the singular and the polynomial cases can also be seen from the any of the four Appell functions. For example, the same conditions for the singular cases of AppellF1 can be seen entering AppellF1:-Singularities(), so with no arguments, and in the same way the conditions for 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;

(10)

Single Power Series with Hypergeometric Coefficients

  

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 AppellF1 and AppellF3,

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

(11)

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

(12)
  

As indicated in the formulas above, for these two Appell functions 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 and 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, for AppellF1 the convergent case at z1&equals;1 requires that 0<Rec&plus;a&plus;b1.

  

The situation is different for AppellF2 and AppellF4, where 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:

FunctionAdvisorsum_form&comma;AppellF2

F2a&comma;b__1&comma;b__2&comma;c__1&comma;c__2&comma;z__1&comma;z__2=m=0n=0am+nb__1mb__2nz__1mz__2nc__1mc__2nm!n!&comma;z__2+z__1<1,F2a&comma;b__1&comma;b__2&comma;c__1&comma;c__2&comma;z__1&comma;z__2=k=0akb__1kF12a+k,b__2;c__2;z__2z__1kc__1kk!&comma;z__2+z__1<1,F2a&comma;b__1&comma;b__2&comma;c__1&comma;c__2&comma;z__1&comma;z__2=k=0akb__2kF12a+k,b__1;c__1;z__1z__2kc__2kk!&comma;z__2+z__1<1

(13)

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

(14)
  

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.

Analytic Extension from the Appell Series to the Appell Functions

  

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 AppellF3

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;uz__1uc+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;uz__1F12a__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+1uz__1+1a__1vz__2+1a__2&DifferentialD;u&DifferentialD;vΓb__1Γb__2Γcb__1b__2&comma;0<b__10<b__20<c+b__1+b__2

(15)
  

In the case of AppellF4, single integral representation exists only for particular values of the function's parameters, for example two cases are

FunctionAdvisorintegral_form&comma;AppellF41..2

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

(16)
  

All these integral representations are the starting point for the derivation of many of the identities known for the four Appell functions.

Euler-Type and Contiguity Identities

  

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,

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&semi;

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

(17)
  

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

eval&comma; z__1 &equals; 1&semi;

F4