This presentation is a continuation of the reconsideration of solutions to second-order linear differential equations with polynomial coefficients. The general form of the equation is

$A_2\left(z\right)\frac{d^{2}F}{d{z}^2}+A_1\left(z\right)\frac{dF}{dz}+A_0\left(z\right)F=0A_i=\sum _k{a}_{ik}{z}^k$

Solutions to cases of this equation are controlled by the singularity structure of the equation, namely whether singular points exist and what the s-rank of each singular point is. Regular singular points are all assigned an s-rank of unity. The s-rank of an irregular finite singular point is given by

$R\left(z_{\mathrm{s}}\right)=max[s_2-s_1,\frac{s_2-s_0}{2}]$

where s₂, s₁ and s₀ designate the lowest powers of the independent variable with respect to the singularity in each coefficient polynomial. The s-rank of an irregular singular point at infinity is given by

$R\left(\infty \right)=max[r_1-r_2+2,\frac{r_0-r_2+4}{2}]$

where r₂, r₁ and r₀ designate the highest powers of the independent variable in each coefficient polynomial.

The general process of categorizing cases of the equation can be extended to coefficient polynomials of arbitrary degree, but in this part of the presentation will be limited to constant, linear or quadratic polynomials. As seen at the end of the first part of the presentation, a quadratic coefficient on the first derivative of the equation is already enough complexity to go beyond functions in common use in physics.

It should be noted that the two coefficient polynomials

$a_0+a_{1}z+a_2{z}^2{a}_0+a_2{z}^2$

make the same contribution to the values of s-ranks, since the formulae for the values depend on the lowest and highest powers at the singularities. That means that solutions to equations with coefficients of the second form can always be found from those with coefficients of the first form by setting a₁ equal to zero.

A tabulation of the singularity structure of each case of constant, linear and quadratic coefficients is available here. The term “multisymbol” denotes the collection of s-ranks for all singular points. They are first given by increasing complexity of the coefficients, from constant on the second derivative through quadratics on all derivatives. The multisymbols are then grouped by identical symbols: these equations will all have a similar solution.

The tabulation includes sets of transformed multisymbols, for a change to a squared variable, a cubed variable, a square root variable, a cube root variable and a cubed square root variable. Three of these are transformations considered in another presentation. These three transformations will apply to only one finite singular point at a time, leaving any others unaffected.

The remaining transformations are simple enough to describe that no tabulation is needed. Splitting off a power from the solutions will at most add a regular singular point at either a finite location or infinity, but not change the singularity structure otherwise. Splitting off an exponential of the form $e^{z^k}$ will not change finite singular points, but will add an irregular singular point at infinity of s-rank k+1 unless there is already something equivalent or worse there.

Keep in mind when using this tabulation that the exponents s₁ and s₀ represent factors with respect to the finite singularity. If the polynomial on the second derivative has a leading constant term, then there will be one finite singular point not at the origin for a linear polynomial, and one or two singular points not at the origin for a quadratic polynomial. The exponents for the other coefficients are in that case meant with respect to locations of these finite singular points not at the origin.

Also keep in mind that discriminants are assumed to be nonzero, so that any factored quadratic coefficients will have distinct roots. Cases of zero discriminant need to be addressed separately, and are not included in the tabulation.

The tabulation is useful in several ways. It first shows the number of different types of special functions that can occur in solutions to these differential equations, where each unique multisymbol corresponds to a different function. After identifying the multisymbols for known special functions, it is then simple to know which equations require functions more complicated than usually encountered for their general solutions. The sets of transformed multisymbols indicate how a change of variable can cast a given equation in terms of a known equation, and explain the connections previously found between solutions.

First consider identification of multisymbols in terms of known functions. It was pointed out in the first part of the presentation that the second-order equation with constant coefficients

$a_{20}\frac{d^{2}F}{d{z}^2}+a_{10}\frac{dF}{dz}+a_{00}F=0$

has a multisymbol { ; 2 } corresponding to a general solution in terms of exponentials of the independent variable. Looking at the grouped multisymbols, only three other equations have this multisymbol: those multiplied by a single or double power of the variable at the singularity. Since the powers can be factored out and canceled, these are literally the same equation with the same solution. The Euler equation

$a_{22}{z}^2\frac{d^{2}F}{d{z}^2}+a_{11}z\frac{dF}{dz}+a_{00}F=0$

has the multisymbol { 1 ; 1 } corresponding to a general solution in terms of powers of the independent variable. This is the only equation in the set with this multisymbol.

Now consider three kinds of hypergeometric functions: Gauss, confluent and reduced confluent. The differential equations for these three functions are

$z(1-z)\frac{d^{2}F}{d{z}^2}+[c-(a+b+1\left)z\right]\frac{dF}{dz}-abF=0$

$z\frac{d^{2}F}{d{z}^2}+(c-z)\frac{dF}{dz}-aF=0$

$z\frac{d^{2}F}{d{z}^2}+c\frac{dF}{dz}-F=0$

and the solutions are denoted by ${}_{2}F_1(a,b;c;z)$ , ${}_{1}F_1(a,c;z)$ and ${}_{0}F_1(c;z)$ , respectively. A second linearly independent solution to each equation is found by setting $F\left(z\right)=z^{k}f\left(z\right)$ in each equation, choosing k to remove a resulting coefficient on each third term inverse in the independent variable, and identifying the new parameters for each function. The result is that these equations have similar pairs of linearly independent functions:

${}_{2}F_1(a,b;c;z)z^{1-c}{}_{2}F_1(a-c+1,b-c+1;2-c;z)$

${}_{1}F_1(a,c;z)z^{1-c}{}_{1}F_1(a-c+1,2-c;z)$

${}_{0}F_1(c;z)z^{1-c}{}_{0}F_1(2-c;z)$

The equation for the Gauss hypergeometric function has regular singular points at zero and one, as well as a regular singular point at infinity, and thus has the multisymbol { 1 , 1 ; 1 }. There are several ways to realize the equation among the grouped multisymbols, corresponding to whether the first derivative has constant or linear parts for its coefficient. This corresponds to independent constants on these respective powers in the Gauss hypergeometric equation.

The equation for the confluent hypergeometric function has a regular singular point at the origin and an irregular singular point at infinity, with the multisymbol { 1 ; 2 }. There appear to be two different ways to realize this equation in the grouped list: with two or three linear coefficients or with two or three quadratic coefficients. A closer look at the latter set shows that whenever there is a quadratic in equations with this multisymbol, there is at least a linear on the other derivatives, which means that a linear term can be canceled across the entire equation. The only way to realize this multisymbol is with a linear on the second derivative and at least one more linear on either term, and this explains the grouping

{ L , L , C } , { L , C , L } and { L , L , L }

of similar solutions identified in the previous presentation. The general solution includes in each case one part with an explicit power as a factor, but since the origin is already a regular singular point the process of splitting off a power does not change the singularity structure. The various solutions also have different exponential factors, but since the singular point at infinity is already irregular with the same s-rank as an exponential function the process of splitting off an exponential does not change the singularity structure either.

This last statement has a direct implication: the confluent hypergeometric function has a simple identity that does not occur for the other hypergeometric functions. Setting $F\left(z\right)=e^{z}f\left(z\right)$ its equation becomes

$z\frac{d^{2}f}{d{z}^2}+(c+z)\frac{df}{dz}+(c-a)f=0$

The solution of this last equation is the basis for an identity that allows one to change the sign of the independent variable in this function by multiplying by an exponential:

$e^z{}_{1}F_1(b,c;-z)={}_{1}F_1(c-b,c;z)$

This identity occurs because the coefficients of the first and second terms of the equation are both linear, and splitting off the exponential does not alter the form of the equation. Another way of saying this is that splitting off the exponential does not alter the singularity structure, since the singular point at infinity already has the same s-rank as the exponential.

There is another special relationship between the confluent hypergeometric function and the exponential: when the two parameters of the former are equal, its differential equation becomes

$z\frac{d^{2}F}{d{z}^2}+(c-z)\frac{dF}{dz}-cF=z^{1-c}\frac{d}{dz}\left[z^c\right(\frac{dF}{dz}-F\left)\right]=0$

and one of the solutions is an exponential itself. This does not mean, however, that the singularity structure of the general solution has changed. The second linearly independent solution is a power and an exponential multiplying an incomplete gamma function, so that the regular singular point at the origin is still there.

While the exponential function shares one equal s-rank with the confluent hypergeometric function, the former does not have a singular point at the origin, and this is why the solution to an equation { C , C , C } with all constant coefficients is distinct in form from the confluent hypergeometric function. A distinction in singularity structure is why the two equations { Q , C , C } and { Q , L , C } also have solutions distinct from the confluent hypergeometric function, namely Gauss hypergeometric functions.

The equation for the reduced confluent hypergeometric function has a regular singular point at the origin and an irregular singular point at infinity, but with the multisymbol { 1 ; 1.5 }. This indicates that it is a different function from the confluent one, yet it turns out that there is a close relationship between the two that explains the solution found for the equation { L , C , C }.

There are two ways to attempt to find the relationship. The process of splitting off an exponential from the reduced confluent hypergeometric function would change its multisymbol to { 1 ; 2 }, and would lead one to think that the reduced confluent equation is a subcase of { L , L , L }. It is in fact not, because comparing both equations shows that the reduced confluent equation corresponds to having both a₁₁ and a₀₁ equal to zero, a subcase that invalidates the solution found before. This explains why certain constants could not simply be set equal to zero in many of the previous solutions: that would alter the singularity structure of the equation, necessitating a different function for its solution.

The apparent relationship just described is a caution that while multisymbols indicate differences between differential equations, there may be circumstances where they do fail to do so. This happens whenever there is a transformation that simplifies the given equation, reducing the s-rank of one or more singular points. In this particular example, one says that an equation with the multisymbol { 1 ; 2 } can be reduced to one with { 1 ; 1.5 } when it is in fact the reduced confluent hypergeometric equation, but not when it is a confluent hypergeometric equation.

The other way to find the relationship is to change to a square root variable, which also alters the multisymbol to { 1 ; 2 }. With $u=\sqrt{z}$ the reduced confluent differential equation becomes

$u\frac{d^{2}F}{d{u}^2}+(2c-u)\frac{dF}{du}-4uF=0$

As for the previous general solution, this equation can be brought into the form of the confluent equation by including an exponential factor. The final result is the identity

${}_{0}F_1(c;z)=e^{-2\sqrt{z}}{}_{1}F_1(c-\frac{1}{2},2c-1;4\sqrt{z})$

An equation of the form { L , C , C } is a reduced confluent hypergeometric equation, but because of this relationship between the two functions the solution can be written as a confluent hypergeometric function of square-root argument. This explains why the solution previously found has this argument, and also why there is difference by a factor of two between the two parameters of the confluent function.

The next group of solutions to understand is

{ C , L , C } , { C , L , L } , { C , C , Q } and { C , L , Q }

The equations all have the multisymbol { ; 3 }, which is why their solutions are similar. This is true even though the last two equations have solutions with exponential factors like $e^{z^2}$ , since splitting off this exponential would set the s-rank of the irregular singular point at infinity to three if it were not already that. These equations are all forms of what is called the biconfluent hypergeometric function in Special Functions: A Unified Theory Based on Singularities, but this does not yet appear to be standard usage. Since this volume is an extremely well-organized enumeration of generalized special functions, following the usage of what is sure to become a standard reference seems prudent.

The biconfluent hypergeometric equation, as given on page 61 of the book,

$\frac{d^{2}F}{d{z}^2}-z\frac{dF}{dz}-aF=0$

also goes by the name of the Hermite differential equation. Its solution, along with solutions of more complicated equations to follow, will be denoted as $F^{\{;3\}}(a;z)$ using a superscripted multisymbol. One could do the same with known functions for complete consistency, and perhaps that will be the future of notation for special functions.

The biconfluent hypergeometric function bears a relationship to the confluent function similar to that between the reduced confluent and confluent, but this time with change to a squared variable that alters the multisymbol to { 1 ; 2 }. With $u=z^2$ the Hermite differential equation becomes

$u\frac{d^{2}F}{d{u}^2}+(\frac{1}{2}-\frac{u}{2})\frac{dF}{du}-\frac{a}{4}F=0$

which is a confluent hypergeometric equation as expected, so that one can make the immediate identification

$F^{\{;3\}}(a;z)={}_{1}F_1(\frac{a}{2},\frac{1}{2};\frac{z^2}{2})$

This relationship explains why the previous general solutions to the equations in this group all have a squared argument in the confluent hypergeometric function, and why all have a parameter of one half. It is now clear why the two constants a₁₁ and a₀₂ cannot both be set equal to zero for the equation { C , L , Q }: the existence of the linear part of the coefficient on the first derivative and the quadratic part of the coefficient on the function each can determine the s-rank of the singular point at infinity, but at least one of them must be present.

Before considering the last case { C , C , L } of second-order equations with linear coefficients, first reconsider the equation

$a_{22}(z+\frac{a_{21}}{2a_{22}}{)}^2\frac{d^{2}F}{d{z}^2}+(a_{10}+a_{11}z)\frac{dF}{dz}+a_{00}F=0$

This equation has a double singularity at $z=-\frac{a_{21}}{2a_{22}}$ where the two previous singularities have converged. This is what is meant by a confluent equation: one where singularities flow together. The confluence of the singularities has turned them from two regular singularities into one irregular singularity. There is additionally a regular singularity at infinity, so that the multisymbol is { 2 ; 1 }. The solution previously found has an inverse argument, and this is because a change to an inverse variable alters the multisymbol to { 1 ; 2 }, which is that for a confluent hypergeometric function. Here again additional factors of powers of the independent variable do not change the singularity structure.

Now on to the equation { C , C , L }. This is a case of what Slavyanov and Lay call the reduced biconfluent hypergeometric equation on page 66 of the book. The standard differential equation for this function is

$\frac{d^{2}F}{d{z}^2}-zF=0$

which is more commonly known as the Airy differential equation. The equation has one irregular singular point at infinity and a multisymbol of { ; 2.5 }.

While the Airy functions are well-enough known in physics to count among standard functions, they too have a relationship with confluent hypergeometric functions. A change to a cubed square root variable alters the multisymbol to { 1 ; 2 }, that for a confluent hypergeometric function. With $u=z^{3/2}$ the Airy differential equation becomes

$u\frac{d^{2}F}{d{u}^2}+\frac{1}{3}\frac{dF}{du}-\frac{4u}{9}F=0$

As before this equation can be brought into the form of the confluent equation by including an exponential factor, with the resulting identities

$F^{\{;2.5\}}\left(z\right)=exp(-\frac{2}{3}{z}^{3/2}){}_{1}F_1(\frac{1}{6},\frac{1}{3};\frac{4}{3}{z}^{3/2})={}_{0}F_1(\frac{2}{3};\frac{z^3}{9})$

The outer equality can also be determined directly with a change to an cubed variable by setting $u=z^3$ in the Airy differential equation. The two hypergeometric functions here are linear combinations of standard Airy functions.

These relationships explain the unusual argument in the general solution of the equation { C , C , L }. They also indicate that this general solution can be stated a bit more simply in terms of the reduced confluent hypergeometric function rather than the confluent hypergeometric function.

Since the Airy functions do not have finite singularities with multisymbol { ; 2.5 }, it is useful to understand how they can be directly related to functions that do have singular points at the origin. The reduced confluent hypergeometric function ${}_{0}F_1(\frac{2}{3};w)$ is regular at the origin, but its corresponding linearly independent solution   $w^{1/3}{}_{0}F_1(\frac{4}{3};w)$ has a cube-root branch point there. It is only because the argument of the function is cubed that a regular single power of z appears in the Airy functions, instead of the cube-root branch point. Similarly, the confluent hypergeometric function ${}_{1}F_1(\frac{1}{6},\frac{1}{3};w)$ is regular at the origin, but its corresponding linearly independent solution   $w^{2/3}{}_{1}F_1(\frac{5}{6},\frac{5}{3};w)$ also has a branch point there. Again in this case the power of the argument combines with the power of the branch point to produce a regular single power of z. The identical circumstance occurs for the biconfluent hypergeometric function, so that while ${}_{1}F_1(\frac{a}{2},\frac{1}{2};w)$ is regular at the origin while   $w^{1/2}{}_{1}F_1(\frac{a+1}{2},\frac{3}{2};w)$ has a branch point there, the squared argument combines with the branch point to produce a regular single power of z. Nifty!

Identification of the remaining multisymbols will require the introduction of functions beyond those commonly known. Begin by restating the equation { C , Q , C }:

$a_{20}\frac{d^{2}F}{d{z}^2}+(a_{10}+a_{11}z+a_{12}{z}^2)\frac{dF}{dz}+a_{00}F=0$

This is a case of what Slavyanov and Lay call the triconfluent Heun equation. There is one irregular singular point at infinity and the multisymbol is { ; 4 }. In fact one could consider an apparently more involved equation

$a_{20}\frac{d^{2}F}{d{z}^2}+(a_{10}+a_{11}z+a_{12}{z}^2)\frac{dF}{dz}+(a_{00}+a_{01}z+a_{02}{z}^2+a_{03}{z}^3+a_{04}{z}^4)F=0$

because both of these equations have the same singularity structure. This structure will not be altered by splitting off an exponential with terms up to a cubic, so that the transformation $F\left(z\right)\to exp[\alpha z^3+\beta z^2+\gamma z]F\left(z\right)$ can be used to remove the quartic, cubic and quadratic parts of the coefficient on the third term with appropriate choices of constants. The remaining coefficients in the equation will have different values after the transformation, and the minimal equation to be solved can be written

$\frac{d^{2}F}{d{z}^2}+(b_{10}+b_{11}z+b_{12}{z}^2)\frac{dF}{dz}+(b_{00}+b_{01}z)F=0$

First change the sign of the independent variable with $u=-z$ ,

$\frac{d^{2}F}{d{u}^2}-(b_{10}-b_{11}u+b_{12}{u}^2)\frac{dF}{du}+(b_{00}-b_{01}u)F=0$

then factor the coefficient of the second term:

$b_{10}-b_{11}u+b_{12}{u}^2=b_{12}(u-u_{+})(u-u_{-})u_{\pm }=\frac{b_{11}\pm \sqrt{b_{11}^2-4b_{12}{b}_{10}}}{2b_{12}}$

Now construct a Möbius transformation to move the roots to two as yet unspecified points equidistant from the origin of the independent variable:

$\begin{array}{c}(u-u_{+})(w+w_0)=(u-u_{-})(w-w_0)\\ u=\frac{u_{+}-u_{-}}{2w_0}w+\frac{u_{+}+u_{-}}{2}\end{array}$

The differential equation in terms of the new variable is

$\begin{array}{l}\frac{d^{2}F}{d{w}^2}-b_{12}\frac{(u_{+}-u_{-}{)}^3}{8w_0^3}(w^2-w_0^2)\frac{dF}{dw}\\ +\frac{(u_{+}-u_{-}{)}^2}{4w_0^2}[b_{00}-b_{01}(\frac{u_{+}-u_{-}}{2w_0}w+\frac{u_{+}+u_{-}}{2}\left)\right]F=0\end{array}$

Choosing $w_0=b_{12}^{1/3}\frac{u_{+}-u_{-}}{2}$ to make the overall factor on the second term unity brings this to the form of the triconfluent Heun equation as given on page 104 of Slavyanov and Lay,

$\frac{d^{2}F}{d{z}^2}-(t+z^2)\frac{dF}{dz}+(\lambda -az)F=0$

and a solution to the equation in question is $F^{\{;4\}}[a;t;\lambda ;w(z\left)\right]$ , where in this case

$\begin{array}{c}a=\frac{b_{01}}{b_{12}}t=-\frac{b_{11}^2-4b_{12}{b}_{10}}{2b_{12}^{4/3}}\\ \lambda =b_{12}^{-2/3}(b_{00}-\frac{b_{11}{b}_{01}}{2b_{12}})\\ w\left(z\right)=-b_{12}^{1/3}(z+\frac{b_{11}}{2b_{12}})\end{array}$

The triconfluent Heun function is not yet commonly known in physics. It occurs, for example, in quantum mechanics for the so-called quartic oscillator, when a fourth-order correction term is added to the harmonic oscillator potential. Mathematica does not currently handle Heun functions other than as Mathieu functions, which Slavyanov and Lay call reduced singly confluent Heun functions. The mathematical package Maple does include most Heun functions, although with deficiencies in numerical evaluation.

The triconfluent Heun function provides a solution to the group of equations

{ C , Q , C }, { C , Q , L } and { C , Q , Q }

since they all have the multisymbol { ; 4 }. For the remaining possible second-order equations with coefficient polynomials up to quadratic, the groups and their corresponding solution functions will be listed, but explicit solutions will not be developed. The three equations

{ L , Q , C }, { L , Q , L } and { L , Q , Q }

all have the multisymbol { 1 ; 3 }, and are cases of the biconfluent Heun equation as given on page 102 of Slavyanov and Lay:

$z\frac{d^{2}F}{d{z}^2}+(c-tz-z^2)\frac{dF}{dz}+(\lambda -az)F=0$

The two equations

{ L , C , Q } and { L , L , Q}

both have the multisymbol { 1 ; 2.5 }, and are cases of the reduced biconfluent Heun equation as given on page 106 of Slavyanov and Lay:

$z\frac{d^{2}F}{d{z}^2}+c\frac{dF}{dz}+(\lambda -tz-z^2)F=0$

The five equations

{ Q , C , Q }, { Q , L , Q }, { Q , Q , C }, { Q , Q , L } and { Q , Q , Q }

all have the multisymbol { 1 , 1 ; 2 }, and are cases of the singly confluent Heun equation as given on page 99 of Slavyanov and Lay:

$z(z-1)\frac{d^{2}F}{d{z}^2}+\left[c\right(z-1)-tz(z-1)+dz]\frac{dF}{dz}+(\lambda -taz)F=0$

The two equations

{ Q , C , L } and { Q , L , L }

both have the multisymbol { 1 , 1 ; 1.5 }, and are cases of the reduced singly confluent Heun equation as given on page 106 of Slavyanov and Lay:

$z(z-1)\frac{d^{2}F}{d{z}^2}+\left[c\right(z-1)+dz]\frac{dF}{dz}+(\lambda -tz)F=0$

In enumerating all of the groups, it has been assumed that quadratic coefficient has zero discriminant and thus two distinct roots. The cases of two equal roots are covered by the multisymbols { 2 ; 2 } and { 2 ; 1.5 }. The first of these corresponds to the doubly confluent Heun equation as given on page 104 of Slavyanov and Lay:

$z^2\frac{d^{2}F}{d{z}^2}+(t+cz-z^2)\frac{dF}{dz}+(\lambda -az)F=0$

The multisymbol { 2 ; 1.5 } can be understood by knowing that a change to an inverse variable switches the locations of singularities by not their s-ranks. A typical equation with this multisymbol is

$a_{22}{z}^2\frac{d^{2}F}{d{z}^2}+a_{10}\frac{dF}{dz}+a_{01}zF=0$

Changing to an inverse variable with $z=\frac{1}{w}$ , the equation becomes

$a_{22}{w}^2\frac{d^{2}F}{d{w}^2}+(2a_{22}w-a_{10}{w}^2)\frac{dF}{dw}+\frac{a_{01}}{w}F=0$

The linear part of the coefficient on the second term can be removed by splitting off a power factor with an exponent of minus one-half, which will add constant and linear parts to the coefficient on the third term. After rescaling the independent variable, one will have a case of the reduced doubly confluent Heun equation as given on page 107 of Slavyanov and Lay,

$z^2\frac{d^{2}F}{d{z}^2}-z^2\frac{dF}{dz}+(\lambda -az-\frac{t}{z})F=0$

and the multisymbol { 2 ; 1.5 } corresponds to a reduced doubly confluent Heun function with inverse argument.

The tabulations of transformed multisymbols can be used to indicate the possibility of additional identities, not only among the Heun functions but also between Heun and hypergeometric functions. As with previous identities, a function with fewer arbitrary parameters corresponds to one with fixed values for one or more of the parameters. The relationships among Heun functions are only simple for fixed values of certain parameters, and so are more restricted in this sense than previous identities.

For example, consider a change to a square root variable in the confluent hypergeometric equation, which alters its multisymbol from { 1 ; 2 } to { 1 ; 3 }. With $u=z^{1/2}$ the equation becomes

$u\frac{d^{2}F}{d{u}^2}+(2c-1-2u^2)\frac{dF}{du}-4auF=0$

This is a form of the Heun biconfluent equation with both of the parameters t and λ equal to zero. If the function with all parameters nonzero is notated by $F^{\{1;3\}}(a,c;t;\lambda ;z)$ , then one can write the identity

${}_{1}F_1(a,c;z)=F^{\{1;2\}}(a,c;z)=F^{\{1;3\}}[2a,2c-1;0;0;(2z{)}^{1/2}]$

which means that the confluent hypergeometric function is a case of the biconfluent Heun function with a square root argument. Since the other types of confluent hypergeometric can be written in terms of the confluent function, they can also be written as biconfluent Heun functions with appropriate arguments. In particular, multiplying the biconfluent hypergeometric equation by a single power of the independent variable gives a case of the biconfluent Heun equation, so one can immediately identify

$F^{\{;3\}}(a;z)=F^{\{1;3\}}(a,0;0;0;z)$

which is consistent with the previous identification between the two hypergeometric functions.

Since the confluent hypergeometric function has a simple relation to the biconfluent Heun function, it is expected that the reduced confluent hypergeometric function will have a simple relation to the reduced biconfluent Heun function. Inspecting available transformations, a change to a cube root variable alters the multisymbols appropriately. With $u=z^{1/3}$ the reduced confluent hypergeometric equation becomes

$u\frac{d^{2}F}{d{u}^2}+(3c-2)\frac{dF}{du}-9u^{2}F=0$

which is a form of the reduced biconfluent Heun equation with both of the parameters t and λ equal to zero. With notation as before one can write the identity

${}_{0}F_1(c;z)=F^{\{1;1.5\}}(c;z)=F^{\{1;2.5\}}[3c-2;0;0;(9z{)}^{1/3}]$

The reduced confluent hypergeometric function is related to the confluent hypergeometric function by an exponential and a square root argument. Comparing the identities just found, the reduced biconfluent Heun function will be equal to the product of an exponential with argument of exponent three halves and a biconfluent Heun function with argument of exponent three quarters. With $u=z^{3/4}$ and t = λ = 0 the reduced biconfluent Heun equation becomes

$u\frac{d^{2}F}{d{u}^2}+\frac{(4c-1)}{3}\frac{dF}{du}-\frac{16}{9}{u}^{3}F=0$

For a solution of the form $F\left(u\right)=e^{-k{u}^2}f\left(u\right)$ the equation is then

$u\frac{d^{2}F}{d{u}^2}+[\frac{(4c-1)}{3}-4k{u}^2]\frac{dF}{du}+[4k^2{u}^3-\frac{16}{9}{u}^3-\frac{4(2c+1)}{3}ku]F=0$

and for an appropriate choice of the free constant one has the identity

$F^{\{1;2.5\}}[c;0;0;z]=exp(-\frac{2}{3}{z}^{3/2})F^{\{1;3\}}[\frac{2c+1}{3},\frac{4c-1}{3};0;0;\sqrt{\frac{8}{3}}{z}^{3/4}]$

which is completely consistent with previous identities. If the parameters t and λ were not set equal to zero, then the coefficient of the third term in the reduced biconfluent Heun equation would include $\lambda u^{1/3}\mathrm{and}t{u}^{5/3}$ in addition to the cubic, and there does not appear to be a simple way to transform this full form to the biconfluent Heun equation.

Since the triconfluent Heun equation appears readily enough in physics, it is interesting to develop its identities to the hypergeometric functions and other Heun functions. This will be done for the restricted case t = λ = 0, for which the triconfluent Heun equation is

$\frac{d^{2}F}{d{z}^2}-z^2\frac{dF}{dz}-azF=0$

A change to a cubed variable alters the multisymbol from { ; 4 } to { 1 ; 2 }, that of the confluent hypergeometric function. With $u=z^3$ the equation becomes

$u\frac{d^{2}F}{d{u}^2}+(\frac{2}{3}-\frac{u}{3})\frac{dF}{du}-\frac{a}{9}F=0$

and one has the immediate identity

$F^{\{;4\}}(a;0;0;z)={}_{1}F_1(\frac{a}{3},\frac{2}{3};\frac{z^3}{3})$

Knowing the corresponding linearly independent hypergeometric function indicates that the second linearly independent solution to this case of the triconfluent Heun equation is $z{}_{1}F_1(\frac{a+1}{3},\frac{4}{3};\frac{z^3}{3})$ . This solution is regular at the origin as expected from the multisymbol.

The confluent hypergeometric function in this identity can represent a reduced confluent hypergeometric function when the first parameter is half of the second. With the restriction a = 1 the identity between the two hypergeometric functions allows one to write

$F^{\{;4\}}(1;0;0;z)=exp\left(\frac{z^3}{6}\right){}_{0}F_1(\frac{5}{6};\frac{z^6}{144})$

The reduced confluent hypergeometric function can represent an Airy function when its parameter is equal to two thirds or four thirds, which is not the case here. The triconfluent Heun function apparently cannot be directly related to Airy functions.

Finally, consider relationships of the triconfluent Heun function to the biconfluent and reduced biconfluent Heun functions, again all for t = λ = 0. A change to a cubed square root variable alters the triconfluent multisymbol from { ; 4 } to { 1 ; 3 }, that of the biconfluent function. With $u=z^{3/2}$ the equation becomes

$u\frac{d^{2}F}{d{u}^2}+(\frac{1}{3}-\frac{2}{3}{u}^2)\frac{dF}{du}-\frac{4}{9}auF=0$

and the identity between the two functions is

$F^{\{;4\}}[a;0;0;z]=F^{\{1;3\}}[\frac{2}{3}a,\frac{1}{3};0;0;\sqrt{\frac{2}{3}}{z}^{3/2}]$

The biconfluent Heun function here can only represent a reduced biconfluent Heun function when the parameters have the relative values $\frac{2c+1}{3}\mathrm{and}\frac{4c-1}{3}$ from the identity between the two functions, which in this case requires a = 1. The necessity of this value will be clear from the form of the transformed differential equation relating the triconfluent and reduced biconfluent Heun functions.

A change to a squared variable alters the triconfluent multisymbol from { ; 4 } to { 1 ; 2.5 }, that of the reduced biconfluent function. First remove the first-derivative term of the triconfluent Heun equation by splitting off a cubed exponential. With $F\left(z\right)=e^{z^3/6}f\left(z\right)$ the equation becomes

$\frac{d^{2}f}{d{z}^2}+[(1-a)z-\frac{1}{4}{z}^4]f=0$

Now with $u=z^2$ the equation is

$u\frac{d^{2}f}{d{u}^2}+\frac{1}{2}\frac{df}{du}+\frac{1}{4}[(1-a)\sqrt{z}-\frac{1}{4}{u}^2]f=0$

and it is clear that a = 1 removes the square root term. The restricted identity between the two functions is

$F^{\{;4\}}[1;0;0;z]=exp\left(\frac{z^3}{6}\right)F^{\{1;2.5\}}[\frac{1}{2};0;0;\frac{z^2}{{16}^{1/3}}]$

which is completely consistent with the identity between the biconfluent and reduced biconfluent Heun functions.

Summary table of distinct multisymbols for second-order differential equations with polynomial coefficients up to quadratic and their corresponding general solution functions:

The solutions can be accompanied by powers and exponentials that do not alter the singularity structure.

Uploaded 2013.07.31 — Updated 2013.08.07 analyticphysics.com