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
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:
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,b__1,b__2,c,z__1,z__2=∑_k1=0∞∑_k2=0∞a_k1+_k2b__1_k1b__2_k2z__1_k1z__2_k2c_k1+_k2_k1!_k2!,z__1<1∧z__2<1
From this definition, by swapping the AppellF1 variables subscripted with the numbers 1 and 2, the function remains the same; hence
FunctionAdvisorsymmetries,AppellF1
F1a,b__2,b__1,c,z__2,z__1=F1a,b__1,b__2,c,z__1,z__2
Analogously, the definition and symmetry of AppellF2 are
FunctionAdvisordefinition,AppellF2
F2a,b__1,b__2,c__1,c__2,z__1,z__2=∑_k1=0∞∑_k2=0∞a_k1+_k2b__1_k1b__2_k2z__1_k1z__2_k2c__1_k1c__2_k2_k1!_k2!,z__1+z__2<1
FunctionAdvisorsymmetries,AppellF2
F2a,b__2,b__1,c__2,c__1,z__2,z__1=F2a,b__1,b__2,c__1,c__2,z__1,z__2
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,AppellF3
F3a__1,a__2,b__1,b__2,c,z__1,z__2=∑_k1=0∞∑_k2=0∞a__1_k1a__2_k2b__1_k1b__2_k2z__1_k1z__2_k2c_k1+_k2_k1!_k2!,z__1<1∧z__2<1
FunctionAdvisorsymmetries,AppellF3
F3a__2,a__1,b__2,b__1,c,z__2,z__1=F3a__1,a__2,b__1,b__2,c,z__1,z__2,F3b__1,a__2,a__1,b__2,c,z__1,z__2=F3a__1,a__2,b__1,b__2,c,z__1,z__2,F3a__1,b__2,b__1,a__2,c,z__1,z__2=F3a__1,a__2,b__1,b__2,c,z__1,z__2
FunctionAdvisordefinition,AppellF4
F4a,b,c__1,c__2,z__1,z__2=∑_k1=0∞∑_k2=0∞a_k1+_k2b_k1+_k2z__1_k1z__2_k2c__1_k1c__2_k2_k1!_k2!,z__1+z__2<1
FunctionAdvisorsymmetries,AppellF4
F4a,b,c__2,c__1,z__2,z__1=F4a,b,c__1,c__2,z__1,z__2,F4b,a,c__1,c__2,z__1,z__2=F4a,b,c__1,c__2,z__1,z__2
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,AppellF1
F1a,b__1,b__2,c,z__1,z__2,c::ℤ0,−∧a::ℤ0,−∧b__1::¬ℤ0,−∧a<c∨c::ℤ0,−∧a::ℤ0,−∧b__2::¬ℤ0,−∧a<c∨c::ℤ0,−∧a::ℤ0,−∧b__1::ℤ0,−∧b__2::ℤ0,−∧a<c∧b__1+b__2<c∨c::ℤ0,−∧a::¬ℤ0,−∧b__1::¬ℤ0,−∨c::ℤ0,−∧a::¬ℤ0,−∧b__2::¬ℤ0,−∨c::ℤ0,−∧a::¬ℤ0,−∧b__1::ℤ0,−∧b__2::ℤ0,−∧b__1+b__2<c
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,b1,b2,c,z1,z2↦a::ℤ0,−,c::¬ℤ0,−,a::ℤ0,−,c::ℤ0,−,c≤a,b1::ℤ0,−,b2::ℤ0,−,c::¬ℤ0,−,b1::ℤ0,−,b2::ℤ0,−,c::ℤ0,−,c≤b1+b2,b1::ℤ0,−,c::¬ℤ0,−,b2::ℤ0,−,c::¬ℤ0,−
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,AppellF1
F1a,b__1,b__2,c,z__1,z__2=∑m=0∞∑n=0∞am+nb__1mb__2nz__1mz__2ncm+nm!n!,z__1<1∧z__2<1,F1a,b__1,b__2,c,z__1,z__2=∑k=0∞akb__1kF12a+k,b__2;c+k;z__2z__1kckk!,z__1<1,F1a,b__1,b__2,c,z__1,z__2=∑k=0∞akb__2kF12a+k,b__1;c+k;z__1z__2kckk!,z__2<1
FunctionAdvisorsum_form,AppellF3
F3a__1,a__2,b__1,b__2,c,z__1,z__2=∑m=0∞∑n=0∞a__1ma__2nb__1mb__2nz__1mz__2ncm+nm!n!,z__1<1∧z__2<1,F3a__1,a__2,b__1,b__2,c,z__1,z__2=∑k=0∞a__1kb__1kF12a__2,b__2;c+k;z__2z__1kckk!,z__1<1,F3a__1,a__2,b__1,b__2,c,z__1,z__2=∑k=0∞a__2kb__2kF12a__1,b__1;c+k;z__1z__2kckk!,z__2<1
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=1 requires that 0<−Re−c+a+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,AppellF2
F2a,b__1,b__2,c__1,c__2,z__1,z__2=∑m=0∞∑n=0∞am+nb__1mb__2nz__1mz__2nc__1mc__2nm!n!,z__2+z__1<1,F2a,b__1,b__2,c__1,c__2,z__1,z__2=∑k=0∞akb__1kF12a+k,b__2;c__2;z__2z__1kc__1kk!,z__2+z__1<1,F2a,b__1,b__2,c__1,c__2,z__1,z__2=∑k=0∞akb__2kF12a+k,b__1;c__1;z__1z__2kc__2kk!,z__2+z__1<1
FunctionAdvisorsum_form,AppellF4
F4a,b,c__1,c__2,z__1,z__2=∑m=0∞∑n=0∞am+nbm+nz__1mz__2nc__1mc__2nm!n!,z__2+z__1<1,F4a,b,c__1,c__2,z__1,z__2=∑k=0∞akbkF12a+k,b+k;c__2;z__2z__1kc__1kk!,z__2+z__1<1,F4a,b,c__1,c__2,z__1,z__2=∑k=0∞akbkF12a+k,b+k;c__1;z__1z__2kc__2kk!,z__2+z__1<1
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.
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,AppellF3
F3a__1,a__2,b__1,b__2,c,z__1,z__2=Γc∫011−u−1+b__1F12a__2,b__2;c−b__1;z__2uu−c+b__1+11+u−1z__1a__1ⅆuΓb__1Γc−b__1,0<ℜb__1∧0<ℜc∧0<−ℜ−c+b__1,F3a__1,a__2,b__1,b__2,c,z__1,z__2=Γc∫011−ub__2−1F12a__1,b__1;c−b__2;uz__1u−c+b__2+11+u−1z__2a__2ⅆuΓb__2Γc−b__2,0<ℜb__2∧0<ℜc∧0<−ℜ−c+b__2,F3a__1,a__2,b__1,b__2,c,z__1,z__2=Γc∫01uρ−11−uc−ρ−1F12a__1,b__1;ρ;uz__1F12a__2,b__2;c−ρ;−u−1z__2ⅆuΓρΓc−ρ,0<ℜc,F3a__1,a__2,b__1,b__2,c,z__1,z__2=Γc∫01∫01−vu−1+b__1vb__2−11−u−v−c+b__1+b__2+1−uz__1+1a__1−vz__2+1a__2ⅆuⅆvΓb__1Γb__2Γc−b__1−b__2,0<ℜb__1∧0<ℜb__2∧0<−ℜ−c+b__1+b__2
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,AppellF41..2
F4a,b,c__1,c__2,z__1,z__2=Γc__1∫01ub−1F12a2,12+a2;c__1;4u2z__1z__2−1+z__1+z__2u21−ub−c__1+11+−z__1−z__2uaⅆuΓbΓc__1−b,z__1+z__2<1∧c__1=c__2∧0<ℜb∧0<ℜc__1∧0<−ℜ−c__1+b∨z__1+z__2<1∧c__1=c__2∧0<ℜb∧0<ℜc__2∧0<−ℜ−c__2+b,F4a,b,c__1,c__2,z__1,z__2=∫0∞u2a−1F10;c__1;z__1u24F10;c__2;z__2u24ⅇuⅆuΓ2a,z__1+z__2<1∧b=a+12∧0<ℜa∧ℜz__1+z__2<1∧ℜz__1−z__2<1∧−ℜz__1−z__2<1∧−ℜz__1+z__2<1
All these integral representations are the starting point for the derivation of many of the identities known for the four Appell functions.
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,a−b,c2−a is a non-positive integer,
AppellF4a, b, c__1, c__2, z__1, z__2 = AppellF4:-TransformationsEuler1a, b, c__1, c__2, z__1, z__2;
F4a,b,c__1,c__2,z__1,z__2=Γc__2Γb−a−z__2−aF4a,a−c__2+1,a−b+1,c__1,1z__2,z__1z__2Γc__2−aΓb+Γc__2Γa−b−z__2−bF4b,1+b−c__2,b−a+1,c__1,1z__2,z__1z__2Γc__2−bΓa
and this identity can be used to evaluate AppellF4 at z1=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, z__1 = 1;
F4a,b,c__1,c__2,1,z__2=Γc__2Γb−a−z__2−aF34a,a−c__2+1,a2−b2+c__12,a2−b2+12+c__12;c__1,a−b+1,a−b+c__1;4z__2Γc__2−aΓb+Γc__2Γa−b−z__2−bF34b,1+b−c__2,b2−a2+c__12,b2−a2+12+c__12;c__1,b−a+1,b−a+c__1;4z__2Γc__2−bΓa
A contiguity transformation for AppellF4
AppellF4a, b, c__1, c__2, z__1, z__2 = AppellF4:-TransformationsContiguity2a, b, c__1, c__2, z__1, z__2;
F4a,b,c__1,c__2,z__1,z__2=F4b,a+n,c__1,c__2,z__1,z__2−bz__1∑k=1nF4a+k,b+1,c__1+1,c__2,z__1,z__2c__1−bz__2∑k=1nF4a+k,b+1,c__1,c__2+1,z__1,z__2c__2
Each of the four Appell functions satisfy a linear system of partial differential equations, for example for AppellF1
FunctionAdvisorDE,AppellF1
fa,b__1,b__2,c,z__1,z__2=F1a,b__1,b__2,c,z__1,z__2,∂2∂z__12fa,b__1,b__2,c,z__1,z__2=−z__2∂2∂z__1∂z__2fa,b__1,b__2,c,z__1,z__2z__1+−a−b__1−1z__1+c∂∂z__1fa,b__1,b__2,c,z__1,z__2z__1z__1−1−b__1z__2∂∂z__2fa,b__1,b__2,c,z__1,z__2z__1z__1−1−fa,b__1,b__2,c,z__1,z__2ab__1z__1z__1−1,∂2∂z__1∂z__2fa,b__1,b__2,c,z__1,z__2=−z__2∂2∂z__22fa,b__1,b__2,c,z__1,z__2z__1−b__2∂∂z__1fa,b__1,b__2,c,z__1,z__2z__2−1+−a−b__2−1z__2+c∂∂z__2fa,b__1,b__2,c,z__1,z__2z__1z__2−1−fa,b__1,b__2,c,z__1,z__2ab__2z__1z__2−1
By using differential algebra techniques, this PDE system, as well as the ones corresponding to each of the other Appell functions, 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, z__2 = AppellF1a, b__1, b__2, c, z__1, z__2;
F1z__1,z__2=F1a,b__1,b__2,c,z__1,z__2
simplifyop1, 2, PDEtools:-casesplitPDEtools:-dpolyform, no_Fn, lexF1, ivars = z__1, z__2, diffalg
∂3∂z__23F1z__1,z__2=a+2b__2+4z__22+−a+b__1−b__2−3z__1−c−b__2−2z__2+z__1c−b__1+1∂2∂z__22F1z__1,z__2+b__2+12a+b__2+2z__2+−a+b__1−1z__1−c∂∂z__2F1z__1,z__2+F1z__1,z__2ab__2z__2z__2−1−z__2+z__1
This is a linear ODE with four regular singularities, one of which is located at z1
DEtoolssingularitiessubsF1z__1,z__2=F1z__2,
regular=0,1,z__1,∞,irregular=∅
When applying the same procedure to the other Appell functions, the result is a fourth order linear ODE with singularities of increasing complexity. The singularities of those fourth order linear ODES behind AppellF2, AppellF3 and AppellF4 can be viewed directly using the Singularities command of the MathematicalFunctions:-Evalf package; for instance for AppellF4 the singularities of the underlying ODE are
MathematicalFunctions:-Evalf:-SingularitiesAppellF4a, b, c__1, c__2, z__1, z__2;
0,z__1−1a+b−c__1+1a+b−c__1−2c__2+3c__1−1−b+a−c__1+1−b+a,z__1+1−2z__1,z__1+1+2z__1,∞+∞I
[1] Appell, P.; Kampe de Feriet, J. Fonctions Hypergeometriques et hyperspheriques. Gauthier-Villars, 1926.
[2] Srivastava, H. M.; Karlsson, P. W. Multiple Gaussian Hypergeometric Series. Ellis Horwood, 1985.
The AppellF1, AppellF2, AppellF3, and AppellF4 commands were introduced in Maple 2017.
For more information on Maple 2017 changes, see Updates in Maple 2017.
See Also
AppellF1
AppellF2
AppellF3
AppellF4
FunctionAdvisor
Heun
hypergeom
MathematicalFunctions
MeijerG
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