Split-Quaternion Higher Special Functions

Introduction

This article treats the higher special functions of a split-quaternion variable. It states the general principle by which a classical special function extends to the algebra, defines the Gamma function, the Bessel functions, the hypergeometric function and the orthogonal polynomials in that framework, describes the special functions coming from the commutative subalgebras and those coming from the analysis, and records the conventions used.

The split-quaternion algebra, its involution, its idempotents and its null basis are assumed from Split-Quaternion Algebra; the units and the three-way classification from Split-Quaternion Norm and Invertibility; the nilpotents from Split-Quaternion Zero Divisors; the exponential and the elementary functions from Split-Quaternion Elementary Functions; and the convergence of power series from Split-Quaternion Analysis, §Power Series and Analytic Functions. The functions of matrices are those of Holomorphic Functional Calculus, the special functions of the division-algebra case those of Quaternion Special Functions, and of the two-dimensional hyperbolic case those of Split-Complex Special Functions. Nothing physical is invoked.

The General Principle

Theorem (Extension by Substitution). Let $F$ be a classical special function given by a power series

$$ F(A) = \sum_{n\geq 0} a_n A^n $$

with real coefficients $a_n$ and radius of convergence $R$. Then the same series

$$ F(\tilde q) = \sum_{n\geq 0} a_n \tilde q^n $$

converges absolutely for $|\tilde q| < R/\sqrt{2}$ in the Euclidean norm of the algebra, defines a function of the split-quaternion variable, and agrees with $F$ on every commutative subalgebra. If $F$ is entire, so is the extension.

Proof. The bound $|a_n\tilde q^n| \leq (\sqrt{2})^n |a_n|\,|\tilde q|^n$ of Split-Quaternion Analysis, §The Metric Structure, reduces the convergence to the real majorant series.

Theorem (The Reduction on a Plane). If $\tilde q$ lies in a commutative subalgebra $\operatorname{span}\{1,\xi\}$ with $\xi^2 = \pm1$, then

$$ F(\tilde q) = F(s + t\xi) = \tfrac12\big(F(s+t) + F(s-t)\big) + \tfrac12\big(F(s+t) - F(s-t)\big)\,\xi \quad (\xi^2=1), $$

$$ F(\tilde q) = \tfrac12\big(F(s+it) + F(s-it)\big) + \tfrac1{2i}\big(F(s+it) - F(s-it)\big)\,\xi \quad (\xi^2=-1), $$

with the second formula understood through its real series. In particular, in the null coordinates $\tilde q = p\,n_+ + q\,n_-$ of the split-complex plane the value is

$$ F(\tilde q) = F(p)\, n_+ + F(q)\, n_- . $$

Proof. In the split case the subalgebra is $\mathbb{R}\times\mathbb{R}$ through the null basis, and $F$ acts on each factor; in the definite case the subalgebra is $\mathbb{C}$ and the displayed formula is the even–odd decomposition.

Theorem (Extension by Interpolation). Let $F$ be a function of one complex variable, holomorphic on a neighbourhood of the roots $\lambda_1, \lambda_2$ of the characteristic polynomial of $\tilde q$; define

$$ F(\tilde q) = \frac{F(\lambda_1) - F(\lambda_2)}{\lambda_1 - \lambda_2}\,\tilde q + \frac{\lambda_1F(\lambda_2) - \lambda_2F(\lambda_1)}{\lambda_1-\lambda_2}, $$

for distinct roots, with the limiting derivative form when the roots coincide. The value depends only on the two roots and on $\tilde q$, and the map is the unique homomorphism on the subalgebra generated by $\tilde q$ taking $\tilde q$ to $F(\tilde q)$.

Proof. The element $\tilde q$ has the spectral decomposition $\tilde q = \lambda_1 P_1 + \lambda_2 P_2$ with the idempotents $P_1 = (\tilde q-\lambda_2)/(\lambda_1-\lambda_2)$ and $P_2 = (\tilde q-\lambda_1)/(\lambda_2-\lambda_1)$, which satisfy $P_1+P_2 = 1$ and $P_1P_2 = 0$; setting $F(\tilde q) = F(\lambda_1)P_1 + F(\lambda_2)P_2$ and collecting terms gives the displayed formula. Uniqueness holds because the values on the two idempotents determine a homomorphism on the two-dimensional subalgebra.

Remark (The Sibling Functions). The error function and the Airy functions of Quaternion Special Functions and Split-Complex Special Functions have real power series, so they too extend by substitution; the Lambert $W$ is defined by the equation $W(\tilde q)e^{W(\tilde q)} = \tilde q$ rather than by a series and is extended by interpolation. The reduction on a commutative subalgebra used throughout is the idempotent principle of Split-Complex Special Functions, §The Idempotent Principle, transported to the split-quaternion variable.

Remark (The Convention Used Here). Where a special function is given by a real power series the extension is by substitution, as in the first theorem, and no convention is needed. Where it is not, the extension used is the interpolation of the last theorem; this is stated explicitly at each use, and it is not claimed to be the only possible extension. In particular the "hypergeometric function of a matrix argument" of random matrix theory, defined through zonal polynomials, is a different convention and is not used here.

The Gamma Function

Definition. On the commutative subalgebras generated by $e_1$ and by $e_2$ the Gamma function is defined by the integral

$$ \Gamma(\tilde q) = \int_0^\infty t^{\tilde q-1} e^{-t}\,\mathrm{d}t , $$

the integral being taken in the subalgebra, and on a general element it is defined by the interpolation of the preceding section.

Theorem (Values and Identities). In the null basis of the split-complex plane, for $p, q > 0$,

$$ \Gamma(p\,n_+ + q\,n_-) = \Gamma(p)\,n_+ + \Gamma(q)\,n_- . $$

For every positive real $t$ the element $t$ is central, so the scalar identities hold in the algebra:

$$ \Gamma(1) = 1, \qquad \Gamma(n+1) = n!, \qquad \Gamma(\tilde q+1) = \tilde q\,\Gamma(\tilde q) \ \text{for central } \tilde q . $$

On the norm-one group the Gamma function of the interpolation is defined off the elements whose characteristic polynomial has a non-positive root, and it agrees with the substitution extension on the elements of scalar square.

Proof. The null-basis formula follows from $t^{\tilde q} = t^p n_+ + t^q n_-$ and the fact that $e^{-t}$ is central; the scalar identities are the classical ones for real arguments, and the last statement is the agreement of the two extensions, which is the theorem on the reduction on a plane.

Remark (No Integral in the General Case). The integral definition does not extend to a general element, because $e^{-t}$ is central but $t^{\tilde q}$ is not defined for a general $\tilde q$ without a branch and a commutative subalgebra; this is the reason for the interpolation convention. Recorded here once and used below.

The Beta Function

Definition. On the same commutative subalgebras the Beta function is

$$ B(p,q) = \int_0^1 t^{p-1}(1-t)^{q-1}\,\mathrm{d}t , $$

the integral being taken in the subalgebra, and it is related to the Gamma function there by the classical identity.

Theorem (Reduction and the Identity). In the null basis of the split-complex plane, for positive real $p,q$,

$$ B(pn_+ + qn_-, r n_+ + s n_-) = B(p,r)\,n_+ + B(q,s)\,n_- , $$

and $B(\tilde q,\tilde p) = \Gamma(\tilde q)\Gamma(\tilde p)/\Gamma(\tilde q+\tilde p)$ for $\tilde q,\tilde p$ in a commutative subalgebra on which both sides are defined; for central arguments the identity holds in the algebra.

Proof. The binary expansion of $t^{\tilde q}$ and $(1-t)^{\tilde p}$ in the null basis is termwise, and $t$ is central, so the integral splits into the two scalar integrals; the classical identity is then applied in each component.

Bessel Functions

Definition. For integer $\nu$ the Bessel function is the entire series

$$ J_\nu(\tilde q) = \sum_{m\geq 0} \frac{(-1)^m}{m!\,\Gamma(m+\nu+1)} \Big(\frac{\tilde q}{2}\Big)^{2m+\nu}, $$

with real coefficients, extended to the algebra by substitution; for non-integer $\nu$ the same series is used on each commutative subalgebra with the corresponding branch of the power.

Theorem (The Reduction). On the split-complex plane, for integer $\nu$,

$$ J_\nu(p\,n_+ + q\,n_-) = J_\nu(p)\,n_+ + J_\nu(q)\,n_- . $$

On an element $v \in V$ with $v^2 = \sigma u^2$, where $u = \sqrt{|v^2|} \geq 0$ and $\sigma = \pm1$ is the sign of the square, put $\xi = v/u$, so that $\xi^2 = \sigma$; then

$$ J_\nu(v) = \begin{cases} \xi^\nu\, I_\nu(u), & \sigma = -1 \ (\text{that is } N(v) > 0),\\ \xi^\nu\, J_\nu(u), & \sigma = +1 \ (\text{that is } N(v) < 0),\end{cases} $$

where $I_\nu$ is the modified Bessel function, the series without the alternating sign. The ordinary and modified Bessel functions are therefore interchanged by a change of the sign of the split-quaternion norm, exactly as the trigonometric and hyperbolic functions are: in the elliptic direction the alternating series produces the monotone function and conversely.

Proof. The series has real coefficients, so the substitution theorem applies. On the split-complex plane the reduction is the null-basis theorem. On $v = u\xi$ one has $v^{2m+\nu} = u^{2m+\nu}\xi^\nu\sigma^m$, so

$$ J_\nu(u\xi) = \xi^\nu \sum_{m\geq0}\frac{(-1)^m\sigma^m}{m!\,\Gamma(m+\nu+1)}\Big(\frac{u}{2}\Big)^{2m+\nu}, $$

and the factor $(-1)^m\sigma^m$ is $1$ when $\sigma = -1$ and $(-1)^m$ when $\sigma = +1$, giving the modified and the ordinary series respectively.

The Hypergeometric Function

Definition. The hypergeometric function is the series

$$ {}_2F_1(a,b;c;\tilde q) = \sum_{n\geq 0}\frac{(a)_n(b)_n}{(c)_n}\frac{\tilde q^n}{n!}, $$

with the rising factorial $(a)_n = a(a+1)\cdots(a+n-1)$, extended to the algebra by substitution whenever $a,b,c$ are real and $c$ is not a non-positive integer, with the radius of convergence of the series.

Theorem (Properties). The extension satisfies the same differential equation with the algebra coefficients,

$$ \tilde q(1-\tilde q)\,{}_2F_1'' + \big(c - (a+b+1)\tilde q\big)\,{}_2F_1' - ab\,{}_2F_1 = 0, $$

where the derivatives are taken in a commutative subalgebra, and the classical transformation and special-value formulas hold for commuting arguments. On the split-complex plane the reduction is ${}_2F_1(a,b;c;pn_+ + qn_-) = {}_2F_1(a,b;c;p)n_+ + {}_2F_1(a,b;c;q)n_-$ whenever the two series converge.

Proof. The coefficients are the classical ones and commute with central elements, so the differential equation is verified series by series; the reduction is the null-basis theorem.

Remark (The Convention). The "hypergeometric function of a matrix argument" of random matrix theory is defined by an expansion in zonal polynomials and is not the function of this section; the two agree only on scalar arguments. This is stated so that no reader identifies the two conventions.

Orthogonal Polynomials

Theorem (Polynomials Extend by Substitution). Let $(P_n)$ be a family of polynomials with real coefficients, such as the Hermite, Legendre or Chebyshev polynomials, defined by a generating function with real coefficients or by a three-term recurrence with real coefficients. Then each $P_n$ extends to the algebra by substitution and satisfies the same recurrence there; the family is determined by the first two terms and the recurrence, and on the split-complex plane the reduction is termwise, $P_n(pn_+ + qn_-) = P_n(p)n_+ + P_n(q)n_-$.

Proof. A real polynomial is a finite real combination of powers, so substitution applies; the three-term recurrence has real coefficients and therefore holds after substitution; the reduction is the null-basis theorem.

Corollary (No Orthogonality Without a Definite Form). The orthogonality relations of the classical polynomials are statements about a positive measure on the line; they do not extend to the algebra, because the algebra has no definite quadratic form and no positive measure supported on the algebra itself. The orthogonality that survives is the one on the definite subalgebras, that is on the plane $\operatorname{span}\{1,e_1\}$ with the definite form, and in the analysis the orthogonality with respect to the invariant measure on the hyperboloids of Split-Quaternion Geometry.

Proof. Orthogonality requires a positive-definite inner product and a measure; the algebra has neither in the general case, and the definite plane and the hyperboloid measures are the two cases where both are available.

The Special Functions from the Matrix Model

Theorem (Functions by Interpolation). Every function holomorphic on a neighbourhood of the two roots of the characteristic polynomial gives a function of the split-quaternion variable by the interpolation formula; the value depends only on the two roots and the element. In particular the square root, the logarithm, the power and the Gamma function are defined on the open sets where the two roots lie in the domain of the branch, and they agree with the extensions by substitution where both are defined.

Proof. The interpolation formula of the first section realises the functional calculus on the two-dimensional subalgebra generated by the element; its uniqueness on the two spectral idempotents is standard, and agreement with the substitution extension holds because the two agree on a power series.

Corollary (The Split-Quaternion Norm of a Function). For a function $F$ defined by interpolation at the roots $\lambda_1,\lambda_2$ of $\tilde q$, the roots of the characteristic polynomial of $F(\tilde q)$ are $F(\lambda_1),F(\lambda_2)$, hence $N(F(\tilde q)) = F(\lambda_1)F(\lambda_2)$; hence the split-quaternion norm of a function can vanish even where the function is defined, and the split-quaternion norm of an algebra-valued function is generally not a function of the split-quaternion norm of its argument. The functions of the algebra for which this happens are exactly the ones whose values at the two eigenvalues have a product that vanishes.

Proof. The interpolation assigns the values $F(\lambda_1),F(\lambda_2)$ to the two spectral idempotents, so these are the roots of the characteristic polynomial of $F(\tilde q)$; the split-quaternion norm is their product, and the vanishing statement is then immediate.

The Special Functions from the Analysis

Theorem (Functions Arising from the Analysis). The analysis of the system produces special functions that are not obtained by substitution from a real series:

the kernel $E_D$ of Split-Quaternion Integration, the fundamental solution of the vector operator, a distribution supported on the light cone and homogeneous of degree $-2$;
the Poisson kernel of the hyperbolic plane, $P(w,\zeta) = \frac{1-|w|^2}{|w-\zeta|^2}$, a function of the two variables of the disc model of Split-Quaternions and Hyperbolic Geometry;
the Legendre functions of the hyperboloid, which are the spherical functions of the pair $\big(\mathrm{SO}^{+}(2,1), SO(2)\big)$ and arise as the eigenfunctions of the invariant Laplacian on the sheets;
the characters of the norm-one group, which are the building blocks of the harmonic analysis of Split-Quaternion Harmonic Analysis.

Proof. The first is the theorem on the fundamental solution of the vector operator in Split-Quaternion Integration; the second is the standard Poisson kernel of the disc, whose invariance is the equivariance of Split-Quaternions and Hyperbolic Geometry; the third and fourth are the standard harmonic analysis of the symmetric space $\mathrm{SO}^{+}(2,1)/SO(2)$ and of its group, treated in Harmonic Analysis on Groups.

Remark (The General Principle). The special functions that extend to the whole algebra by substitution are those given by real power series; the functions that live only on a subalgebra are those defined by an integral or a branch; and the functions that arise from the analysis live on the homogeneous spaces of the isometry group. These three classes are different, and the article has kept them apart.

Summary

A special function given by a real power series extends to the split-quaternion algebra by substitution, converges on the ball of the rescaled radius, and on a commutative subalgebra reduces to its values at the two components of the element in the null basis; on an element with scalar square the reduction is by the even and odd parts. The Gamma function is defined by its integral on the commutative subalgebras, where it reduces to the scalar Gamma function in each null component, and by interpolation elsewhere; the Bessel functions and the hypergeometric function are defined by their real series and reduce on the subalgebras; the orthogonal polynomials extend by substitution, but their orthogonality does not, because the algebra has no definite form and no positive measure.

The interpolation gives a second, general extension, computed from the two roots of the characteristic polynomial, and it agrees with the substitution extension where both are defined. The special functions that arise from the analysis are of a third kind: the fundamental-solution kernel supported on the light cone, the Poisson kernel of the hyperbolic plane, the Legendre functions of the hyperboloid and the characters of the norm-one group. Every convention used in this article is stated at the point of use, and the "matrix-argument" hypergeometric function of random matrix theory is explicitly not the one used here.

Summary of Notation

Symbol	Meaning	Article
substitution extension	$F(\tilde q) = \sum a_n\tilde q^n$ for real $a_n$	this article
null-basis reduction	$F(pn_+ + qn_-) = F(p)n_+ + F(q)n_-$	this article
$\Gamma$	the Gamma function, by integral on the subalgebras and by interpolation generally	this article
$J_\nu$	the Bessel function, an entire series for integer $\nu$	this article
${}_2F_1(a,b;c;\tilde q)$	the hypergeometric function, by its series	this article
interpolation formula	the value of a function at the two roots of the characteristic polynomial	this article
$E_D$	the fundamental solution, a function of the analysis	Split-Quaternion Integration
Poisson kernel	$\frac{1-\|w\|^2}{\|w-\zeta\|^2}$ on the disc	Split-Quaternions and Hyperbolic Geometry
Legendre functions	the spherical functions of the hyperboloid	this article
characters	the building blocks of the harmonic analysis	Split-Quaternion Harmonic Analysis