Biquaternion Higher Special Functions

Introduction

This article introduces the higher special functions of a biquaternion variable. It follows Biquaternion Elementary Functions, which defined the exponential, the trigonometric and hyperbolic functions, the logarithm and the power functions, and which established the two regimes: the oscillatory regime ($\theta \neq 0$) and the nilpotent regime ($\theta = 0$), where $\theta = \sqrt{Q_1^2 + Q_2^2 + Q_3^2}$ is the complex norm of the vector part.

The goal is to define the Bessel functions, the orthogonal polynomials, the hypergeometric function, the gamma function and the zeta function, and to establish their basic properties. The article is honest about what is fully understood and what is still open.

Physically these are the functions that appear when a biquaternion wave equation is separated. The Bessel functions are the radial modes of a wave field and the orthogonal polynomials the eigenfunctions of the angular and oscillator problems — Hermite for the oscillator, Legendre for the angular part, Laguerre for the radial hydrogenic part — while the gamma and zeta functions enter only through the definition of the fractional powers and the special values, where the biquaternion extension is largely open. The kernels of the analysis — the Cauchy kernel, the Poisson kernel and the Green's function — are the propagators and boundary kernels of the field, the objects through which the analysis of Biquaternion Integration becomes a calculational tool. The physical readings below are collected from Biquaternion Non Relativistic Quantum Theory, Biquaternion Quantum Fields and Biquaternion Electromagnetism.

The biquaternion algebra $\mathbb{B}$ is assumed from Biquaternion Algebra, together with its scalar-vector decomposition, its biquaternion norm and its four conjugations. The elementary functions are assumed from Biquaternion Elementary Functions, together with the two regimes and the Hamilton polar form. The material and informational sectors $\mathbb{M}_-$ and $\mathbb{M}_+$ are those of The Anti-Hermitian Subspace $\mathbb{M}_-$ as the Material Sector and The Hermitian Subspace $\mathbb{M}_+$ as the Informational Sector.

Throughout, a biquaternion is written

$$ \tilde{Q} = Q_0 e_0 + Q_1 e_1 + Q_2 e_2 + Q_3 e_3, \qquad Q_\mu \in \mathbb{C}, $$

or, more compactly, as

$$ \tilde{Q} = Q_0 e_0 + \mathbf{Q}, \qquad \mathbf{Q} = Q_1 e_1 + Q_2 e_2 + Q_3 e_3. $$

The complex norm of the vector part is $\theta = \sqrt{Q_1^2 + Q_2^2 + Q_3^2}$, and the axis is $\hat{n} = \mathbf{Q}/\theta$ when $\theta \neq 0$.

The higher special functions fall into two classes according to their definition:

• Power-series functions. These are defined by a power series, and the same technique as for the exponential applies: the series is summed in closed form in the two regimes. The Bessel functions, the orthogonal polynomials, and the hypergeometric function belong to this class.
• Integral-defined functions. These are defined by an integral over a real or complex parameter. The extension to biquaternions requires a notion of the integrand for biquaternion arguments, and the convergence and analyticity of the integral are not automatic. The gamma function and the zeta function belong to this class.

The first class is treated in full; the second is treated with the open questions stated honestly.

Bessel Functions

Definition

The Bessel function of the first kind of order $\nu$ is defined by the power series

$$ J_\nu(\tilde{Q}) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m! \, \Gamma(m + \nu + 1)} \left(\frac{\tilde{Q}}{2}\right)^{2m+\nu}. $$

The series converges for every $\tilde{Q} \in \mathbb{B}$, because the Euclidean norm grows at most exponentially with $m$. So $J_\nu$ is an entire function on $\mathbb{B}$ for every $\nu \in \mathbb{C}$.

Reduction to the Vector Part

Write $\tilde{Q} = Q_0 e_0 + \mathbf{Q}$. The scalar part $Q_0 e_0$ commutes with $\mathbf{Q}$, so the powers of $\tilde{Q}$ do not simplify in the same way as for the exponential. The Bessel function of a general biquaternion is therefore not reducible to a simple closed form in terms of the scalar and vector parts.

However, there are two special cases in which the reduction works:

• Pure biquaternions. If $Q_0 = 0$, then $\tilde{Q} = \mathbf{Q}$, and the powers of $\mathbf{Q}$ simplify as in the elementary case.
• Biquaternions with a single direction. If $\tilde{Q} = Q_0 e_0 + \theta \hat{n}$ with $\hat{n}$ fixed, the powers of $\tilde{Q}$ can be computed by the binomial expansion, but the binomial coefficients are biquaternions in general.

We treat the pure case in detail, and we state the general case as an open problem.

The Pure Case, Oscillatory Regime

Let $\tilde{Q} = \mathbf{Q} = \theta \hat{n}$ with $\theta \neq 0$ and $\hat{n}^2 = -e_0$. The powers of $\mathbf{Q}$ satisfy

$$ \mathbf{Q}^{2m+\nu} = \theta^{2m+\nu} \hat{n}^{2m+\nu}. $$

The powers of $\hat{n}$ alternate:

$$ \hat{n}^{2k} = (-1)^k e_0, \qquad \hat{n}^{2k+1} = (-1)^k \hat{n}. $$

Substituting into the series and separating even and odd powers, we obtain

$$ J_\nu(\theta \hat{n}) = \left(\sum_{m=0}^{\infty} \frac{(-1)^m \theta^{2m+\nu}}{m! \, \Gamma(m+\nu+1) 2^{2m+\nu}} \hat{n}^{2m+\nu}\right). $$

The parity of $\hat{n}^{2m+\nu}$ depends on the parity of $\nu$. If $\nu$ is an integer, the sum splits into two cases: $\nu$ even gives only the identity $e_0$, and $\nu$ odd gives only $\hat{n}$. In general, for non-integer $\nu$, the series does not split into two separate series.

For integer $\nu = n$, the result is

$$ J_n(\theta \hat{n}) = \begin{cases} (-1)^{n/2} I_n(\theta) \, e_0 & \text{if } n \text{ is even}, \\ (-1)^{(n-1)/2} I_n(\theta) \, \hat{n} & \text{if } n \text{ is odd}, \end{cases} $$

where $I_n(\theta)$ on the right is the ordinary complex modified Bessel function. The verification is a direct substitution of the alternating powers of $\hat{n}$.

The Pure Case, Nilpotent Regime

Let $\tilde{Q} = \mathbf{Q}$ with $\theta = 0$ and $\mathbf{Q} \neq 0$. Then $\mathbf{Q}^2 = 0$, so $\mathbf{Q}^m = 0$ for all $m \geq 2$. The Bessel series truncates:

$$ J_\nu(\mathbf{Q}) = \frac{1}{\Gamma(\nu+1)} \left(\frac{\mathbf{Q}}{2}\right)^\nu + \frac{(-1)^1}{1! \, \Gamma(\nu+2)} \left(\frac{\mathbf{Q}}{2}\right)^{\nu+2} + \cdots $$

Only the terms with $2m + \nu \leq 1$ contribute. For general $\nu$, the terms with $2m + \nu \geq 2$ vanish. So the series reduces to at most two terms, depending on the value of $\nu$. For integer $\nu = n \geq 0$, the series reduces to

$$ J_n(\mathbf{Q}) = \begin{cases} \frac{1}{n!} \left(\frac{\mathbf{Q}}{2}\right)^n & \text{if } n = 0 \text{ or } n = 1, \\ 0 & \text{if } n \geq 2, \end{cases} $$

where for $n = 0$ the term is $e_0$ and for $n = 1$ the term is $\mathbf{Q}/2$. For non-integer $\nu$, the series reduces to the terms with $2m + \nu < 2$.

The General Case

For a general biquaternion $\tilde{Q} = Q_0 e_0 + \mathbf{Q}$ with both $Q_0 \neq 0$ and $\mathbf{Q} \neq 0$, the Bessel function does not reduce to a closed form in terms of elementary functions. The powers of $\tilde{Q}$ involve the non-commutative products of $Q_0 e_0$ and $\mathbf{Q}$, and the series is a genuine biquaternion series.

The general case is open. The natural approach would be to use the Hamilton polar form $\tilde{Q} = e^{Q_0} \exp(\theta \hat{n})$ and to express $J_\nu(\tilde{Q})$ in terms of $J_\nu$ evaluated at the polar factors, but the functional equation for the Bessel function under multiplication does not hold in general, because the Bessel function is not multiplicative.

Modified Bessel Functions

The modified Bessel function of the first kind is defined by

$$ I_\nu(\tilde{Q}) = \sum_{m=0}^{\infty} \frac{1}{m! \, \Gamma(m + \nu + 1)} \left(\frac{\tilde{Q}}{2}\right)^{2m+\nu}. $$

The same analysis applies: in the pure oscillatory regime, the result is expressed in terms of the ordinary modified Bessel function; in the pure nilpotent regime, the series truncates.

Physical reading: radial modes. The Bessel functions are the radial modes of a wave: they solve the radial part of the wave equation after separation in a cylindrical or spherical geometry, and an oscillatory argument is the phase factor of an outgoing mode while a modified argument is the exponential decay of an evanescent one. In the framework this separation is the one performed on the spatial directions of the material sector in Biquaternion Non Relativistic Quantum Theory; the pure oscillatory case is the on-shell radial mode and the pure nilpotent case the lightlike limit in which the mode degenerates to its first-order term. The general case, where the scalar part and the vector part are both present, is the off-shell mode, and it is open.

Orthogonal Polynomials

Legendre Polynomials

The Legendre polynomials $P_n(\tilde{Q})$ are defined by the generating function

$$ \frac{1}{\sqrt{1 - 2\tilde{Q} t + t^2}} = \sum_{n=0}^{\infty} P_n(\tilde{Q}) t^n, \qquad |t| < R(\tilde{Q}), $$

where $t$ is a real parameter, the square root is the biquaternion square root (defined via the elementary functions), and the radius $R(\tilde{Q})$ is the distance from the origin to the nearest value of $t$ at which $1 - 2\tilde{Q}t + t^2$ fails to be invertible (for $|\tilde{Q}| \le 1$ one has $R(\tilde{Q}) = 1$, but for $\tilde{Q} = 3e_0$ the quadratic vanishes at $t = 3 \pm 2\sqrt{2}$). Equivalently, they satisfy the recurrence

$$ (n+1) P_{n+1}(\tilde{Q}) = (2n+1) \tilde{Q} P_n(\tilde{Q}) - n P_{n-1}(\tilde{Q}), $$

with $P_0 = e_0$ and $P_1 = \tilde{Q}$.

The Legendre polynomials are polynomials in $\tilde{Q}$ with real coefficients (the coefficients are the ordinary Legendre coefficients, embedded as real scalars). So they are defined for every $\tilde{Q} \in \mathbb{B}$ by the recurrence, and the recurrence involves only the multiplication and addition of biquaternions.

The pure oscillatory case. For $\tilde{Q} = \theta \hat{n}$ with $\theta \neq 0$ and $\hat{n}^2 = -e_0$, the recurrence gives

$$ P_n(\theta \hat{n}) = \begin{cases} P_n(i\theta) \, e_0 & \text{if } n \text{ is even}, \\ \frac{1}{i} P_n(i\theta) \, \hat{n} & \text{if } n \text{ is odd}, \end{cases} $$

where $P_n(i\theta)$ is the ordinary Legendre polynomial evaluated at the complex number $i\theta$. The generator $\hat{n}$ plays the role of the imaginary unit, so no reduction to $P_n(\theta)$ holds: only the even part of $P_n$ survives on $e_0$ and only the odd part on $\hat{n}$. For instance $P_2(\theta\hat{n}) = -(3\theta^2+1)/2$, whereas $P_2(\theta) = (3\theta^2-1)/2$. This follows from the recurrence and the alternating powers of $\hat{n}$.

The pure nilpotent case. For $\tilde{Q} = \mathbf{Q}$ with $\mathbf{Q}^2 = 0$, the recurrence involves powers of $\mathbf{Q}$ that truncate. In particular, $P_n(\mathbf{Q})$ is a polynomial in $\mathbf{Q}$ of degree at most one, because $\mathbf{Q}^2 = 0$. The explicit form is obtained from the recurrence.

Chebyshev Polynomials

The Chebyshev polynomials of the first kind $T_n(\tilde{Q})$ are defined by the recurrence

$$ T_{n+1}(\tilde{Q}) = 2\tilde{Q} T_n(\tilde{Q}) - T_{n-1}(\tilde{Q}), $$

with $T_0 = e_0$ and $T_1 = \tilde{Q}$. The same analysis applies: in the pure oscillatory regime, the polynomials reduce to the ordinary Chebyshev polynomials evaluated at $i\theta$ with $i$ replaced by $\hat{n}$ (so $T_n(\theta\hat{n})$ is $T_n(i\theta)e_0$ or $\frac{1}{i}T_n(i\theta)\hat{n}$ according to the parity of $n$); in the pure nilpotent regime, the polynomials truncate.

Hermite Polynomials

The Hermite polynomials $H_n(\tilde{Q})$ are defined by the recurrence

$$ H_{n+1}(\tilde{Q}) = 2\tilde{Q} H_n(\tilde{Q}) - 2n H_{n-1}(\tilde{Q}), $$

with $H_0 = e_0$ and $H_1 = 2\tilde{Q}$. The same analysis applies.

Laguerre Polynomials

The Laguerre polynomials $L_n(\tilde{Q})$ are defined by the recurrence

$$ (n+1) L_{n+1}(\tilde{Q}) = (2n+1-\tilde{Q}) L_n(\tilde{Q}) - n L_{n-1}(\tilde{Q}), $$

with $L_0 = e_0$ and $L_1 = e_0 - \tilde{Q}$. The same analysis applies.

The General Principle for Polynomials

The orthogonal polynomials are defined by three-term recurrences involving only the multiplication and addition of biquaternions. So they are defined for every $\tilde{Q} \in \mathbb{B}$, and they reduce to the ordinary polynomials in the two pure regimes:

• In the pure oscillatory regime, the polynomial $P_n(\theta \hat{n})$ is the ordinary polynomial evaluated at $i\theta$ with $i$ replaced by $\hat{n}$, i.e. $P_n(i\theta) e_0$ or $\frac{1}{i}P_n(i\theta)\hat{n}$ according to the parity of $n$.
• In the pure nilpotent regime, the polynomial truncates to a polynomial of degree at most one in $\mathbf{Q}$.

The general case (with both $Q_0 \neq 0$ and $\mathbf{Q} \neq 0$) does not reduce to a closed form, because the recurrence involves the non-commutative product $Q_0 \mathbf{Q}$, and the polynomial is a genuine biquaternion polynomial.

Physical reading: eigenfunctions. The orthogonal polynomials are the eigenfunctions of the separated wave equation: Hermite the eigenfunctions of the harmonic oscillator, Legendre the angular eigenfunctions of a central problem, Laguerre the radial eigenfunctions of the hydrogenic problem, Chebyshev the extremal polynomials of approximation. In the framework a stationary quantum state is a solution of the separated equation of Biquaternion Non Relativistic Quantum Theory whose spatial part is one of these polynomials times the corresponding exponential weight, and the biquaternion generalisation is the statement that the same polynomial problem is solved with biquaternion coefficients: in the pure oscillatory regime the polynomial is the ordinary one evaluated at $i\theta$ with $i$ replaced by the axis, and in the pure nilpotent regime it degenerates to its first-order term. The general case is open, as for the Bessel functions.

Hypergeometric Function

Definition

The Gauss hypergeometric function is defined by the power series

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

where $(a)_n = a(a+1) \cdots (a+n-1)$ is the Pochhammer symbol, and $a, b, c \in \mathbb{C}$. The series converges for $\|\tilde{Q}\|_E < 1$.

The Pure Oscillatory Case

For $\tilde{Q} = \theta \hat{n}$ with $\theta \neq 0$ and $\hat{n}^2 = -e_0$, the powers of $\tilde{Q}$ alternate. Separating even and odd powers,

$$ {}_2F_1(a, b; c; \theta \hat{n}) = \left(\sum_{n=0}^{\infty} \frac{(a)_{2n} (b)_{2n}}{(c)_{2n}} \frac{(-1)^n \theta^{2n}}{(2n)!}\right) e_0 + \left(\sum_{n=0}^{\infty} \frac{(a)_{2n+1} (b)_{2n+1}}{(c)_{2n+1}} \frac{(-1)^n \theta^{2n+1}}{(2n+1)!}\right) \hat{n}. $$

The two series are not the ordinary hypergeometric function evaluated at $\theta$, because the Pochhammer symbols are evaluated at $2n$ and $2n+1$, not at $n$. They are the even and odd parts of the hypergeometric function, and they can be expressed in terms of ${}_2F_1$ with different parameters. The precise form is an open problem.

The Pure Nilpotent Case

For $\tilde{Q} = \mathbf{Q}$ with $\mathbf{Q}^2 = 0$, the powers of $\tilde{Q}$ truncate, and the hypergeometric series reduces to

$$ {}_2F_1(a, b; c; \mathbf{Q}) = e_0 + \frac{ab}{c} \mathbf{Q}. $$

The higher terms vanish because $\mathbf{Q}^n = 0$ for $n \geq 2$. This is the analogue of the linear approximation of the hypergeometric function near zero.

Special Cases

The special cases of the hypergeometric function that involve only the elementary functions (the logarithm, the arcsine, the power function) reduce to the elementary functions of a biquaternion variable, which are treated in Biquaternion Elementary Functions. The general hypergeometric function does not reduce to the elementary functions.

The Generalized Hypergeometric Function

The generalized hypergeometric function ${}_pF_q$ is defined by

$$ {}_pF_q(a_1, \dots, a_p; b_1, \dots, b_q; \tilde{Q}) = \sum_{n=0}^{\infty} \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{\tilde{Q}^n}{n!}. $$

The same analysis applies: in the pure oscillatory regime, the series splits into even and odd parts; in the pure nilpotent regime, the series truncates.

Physical reading. The hypergeometric function is the unifying object of the separated problems: the Legendre, Bessel and Hermite functions are special cases of it, so a single biquaternion formula would cover all the modes at once. The framework's use of it is therefore structural rather than numerical: the fact that the power-series methods of Biquaternion Elementary Functions extend to ${}_2F_1$ means the whole family of separated modes is under the same two-regime analysis. The identification of the even and odd parts with the ordinary ${}_2F_1$ at shifted parameters is the open step.

The Gamma Function

The Definition Problem

The gamma function of a complex variable is defined by the integral

$$ \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt, \qquad \operatorname{Re} z > 0. $$

The natural extension to a biquaternion variable would be

$$ \Gamma(\tilde{Q}) = \int_0^\infty t^{\tilde{Q}-1} e^{-t} \, dt, $$

where $t$ is a positive real number, $t^{\tilde{Q}-1} = \exp((\tilde{Q}-1) \log t)$ is the biquaternion power, and $e^{-t}$ is the ordinary real exponential.

The Obstruction

The integral is defined component-wise, but the integrand is a biquaternion-valued function of the real variable $t$, and the convergence of the integral depends on the growth of $t^{\tilde{Q}-1}$ as $t \to 0$ and as $t \to \infty$.

As $t \to 0$. The biquaternion power $t^{\tilde{Q}-1}$ involves $\log t$, which tends to $-\infty$. The behavior depends on the scalar part $Q_0$ and on the vector part $\mathbf{Q}$. In the oscillatory regime, the power is $t^{Q_0-1}(\cos((Q_0-1)\log t \cdot \hat{n}) + \cdots)$, which oscillates and may not converge. In the nilpotent regime, the power is $t^{Q_0-1}(e_0 + (Q_0-1)\log t \cdot \mathbf{Q})$, which converges if $\operatorname{Re} Q_0 > 0$.

As $t \to \infty$. The factor $e^{-t}$ dominates, so the integral converges for all $\tilde{Q}$ with $\operatorname{Re} Q_0 < \infty$.

So the integral converges in the nilpotent regime if $\operatorname{Re} Q_0 > 0$, and its convergence in the oscillatory regime is an open problem.

What Is Known

The gamma function is well-defined in the nilpotent regime for $\operatorname{Re} Q_0 > 0$, and it can be computed in closed form. However, the analytic continuation, the functional equation $\Gamma(\tilde{Q}+1) = \tilde{Q} \Gamma(\tilde{Q})$, and the reflection formula are not established in general. These are open problems.

The Open Questions

1. Convergence. For which biquaternions $\tilde{Q}$ does the integral converge?

2. Functional equation. Does the functional equation $\Gamma(\tilde{Q}+1) = \tilde{Q} \Gamma(\tilde{Q})$ hold? The proof in the complex case uses integration by parts, which requires the boundary terms to vanish. In the biquaternion case, the boundary terms involve the biquaternion power, and their vanishing is not automatic.

3. Analytic continuation. Can the gamma function be continued beyond the region of convergence of the integral?

4. The reflection formula. Does the reflection formula $\Gamma(\tilde{Q}) \Gamma(e_0 - \tilde{Q}) = \pi / \sin(\pi \tilde{Q})$ hold?

These questions are open.

The Zeta Function

The Definition Problem

The Riemann zeta function of a complex variable is defined by the Dirichlet series

$$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \qquad \operatorname{Re} s > 1. $$

The natural extension to a biquaternion variable would be

$$ \zeta(\tilde{Q}) = \sum_{n=1}^{\infty} \frac{1}{n^{\tilde{Q}}}, $$

where $n^{\tilde{Q}} = \exp(\tilde{Q} \log n)$ is the biquaternion power.

The Obstruction

The series is a series of biquaternions. The convergence depends on the behavior of $n^{\tilde{Q}}$ as $n \to \infty$. In the oscillatory regime, $n^{\tilde{Q}}$ involves $\cos(\theta \log n)$ and $\sin(\theta \log n)$, which oscillate. The series may converge conditionally or may not converge at all, depending on $\tilde{Q}$.

In the nilpotent regime, $n^{\tilde{Q}} = n^{Q_0}(e_0 + Q_0 \log n \cdot \mathbf{Q})$, and the series converges absolutely if $\operatorname{Re} Q_0 > 1$.

What Is Known

Almost nothing is known about the biquaternion zeta function. The following are open problems.

The Open Questions

1. Convergence. For which biquaternions $\tilde{Q}$ does the series converge?

2. The Euler product. For positive integer bases the power function is multiplicative: $m^{\tilde{Q}} n^{\tilde{Q}} = \exp(\tilde{Q}\log m)\exp(\tilde{Q}\log n) = (mn)^{\tilde{Q}}$, because the two exponents commute as multiples of $\tilde{Q}$. Hence the Euler product $\prod_p (1 - p^{-\tilde{Q}})^{-1} = \sum_n n^{-\tilde{Q}}$ holds in the region of absolute convergence, and no separate obstruction arises; what remains open is the analytic continuation of the individual factors $p^{-\tilde{Q}}$.

3. Analytic continuation. Can the zeta function be continued beyond the region of convergence?

4. Special values. Are there biquaternion analogues of the special values $\zeta(2) = \pi^2/6$, $\zeta(4) = \pi^4/90$, etc.?

5. The functional equation. Is there a biquaternion analogue of the functional equation relating $\zeta(\tilde{Q})$ to $\zeta(e_0 - \tilde{Q})$?

6. The Riemann hypothesis. Is there an analogue of the Riemann hypothesis?

These questions are entirely open. The biquaternion zeta function is a research problem, not a developed theory.

Physical reading. The gamma and zeta functions enter physics through the analytic continuation used to regularise divergent quantities: the gamma function in the dimensional regularisation of a loop integral, the zeta function in the regularisation of a sum over modes. In the framework both are blocked at the same place, the definition of the biquaternion power for a real argument, which is the logarithm problem of Biquaternion Elementary Functions and of The Wave Equation on $\mathbb{M}_-$. Because the power is defined on the group of units only, the regularised objects exist only away from the null cone, and this is a genuine open problem of the framework rather than a technical gap: the renormalisation of a biquaternion field on the light cone has no gamma-function representation yet.

Special Functions from the Analysis

The Setting

Biquaternion Analysis defined the biquaternion gradient $\tilde{\nabla}$, the d'Alembertian $\Box$ and the convective derivative $\tilde{D}$ on a four-dimensional subspace $V \subset \mathbb{B}$. The solutions of the equations

$$ \tilde{\nabla}\tilde{F} = 0, \qquad \Box\tilde{F} = 0 $$

are the biquaternion analogues of holomorphic functions and harmonic functions. The special functions of the analysis are the kernels and fundamental solutions associated with these equations.

The Cauchy Kernel

The Cauchy kernel is the fundamental solution of the gradient:

$$ \tilde{G}(\tilde{Q}) = \frac{\tilde{Q}^{\natural}}{\|\tilde{Q}\|_E^4}, \qquad \tilde{Q} \neq 0. $$

It satisfies $\tilde{\nabla}\tilde{G} = 2\pi^2 \delta_0 e_0$ in the sense of distributions, where $\delta_0$ is the delta distribution at the origin.

In the oscillatory regime, with $\theta$ real and $\hat{n}$ a root of $-1$, $\tilde{G}(\theta \hat{n}) = -\theta \hat{n} / (\theta^4 \|\hat{n}\|_E^4) = -\hat{n}/(\theta^3 \|\hat{n}\|_E^4)$, which is the quaternion analogue of the Cauchy kernel $1/z$ in complex analysis. The Euclidean norm $\|\hat{n}\|_E$ must be kept: it equals $1$ only on the degenerate branches of the root classification.

In the nilpotent regime, $\tilde{G}(\mathbf{Q}) = -\mathbf{Q}/\|\mathbf{Q}\|_E^4$, which is singular at the origin.

The Poisson Kernel

The Poisson kernel for the ball of radius $r$ is

$$ P(\tilde{Q}, \tilde{Y}) = \frac{r^2 - \|\tilde{Q}\|_E^2}{2\pi^2 r \|\tilde{Q} - \tilde{Y}\|_E^4}, \qquad \|\tilde{Q}\|_E < r, \quad \|\tilde{Y}\|_E = r. $$

It is the kernel that solves the Dirichlet problem for the d'Alembertian on the ball. It is the biquaternion analogue of the Poisson kernel in complex analysis.

The Green's Function

The Green's function for the d'Alembertian on a domain $\Omega$ is the function $G(\tilde{Q}, \tilde{Y})$ that satisfies

$$ \Box_{\tilde{Q}} G(\tilde{Q}, \tilde{Y}) = \delta(\tilde{Q} - \tilde{Y}) e_0, \qquad G(\tilde{Q}, \tilde{Y}) = 0 \text{ for } \tilde{Q} \in \partial \Omega. $$

Its construction is standard from the fundamental solution of the d'Alembertian, and it is the biquaternion analogue of the Green's function in the theory of elliptic partial differential equations.

Physical reading: propagators and boundary kernels. The Cauchy kernel is the Green's function of the massless Dirac operator and the Green's function of the d'Alembertian is the propagator of the wave equation; the Poisson kernel is the boundary-to-interior map of a field on a ball. In the framework these are the objects with which a classical field is computed from its sources in Biquaternion Electromagnetism and Biquaternion Gravity and Cosmology, and with which a quantum amplitude is propagated in Biquaternion Quantum Fields; the potential of a static charge, the retarded field of an accelerating source and the boundary-value solution of Biquaternion Integration are all evaluations of these kernels. The explicit forms in the two regimes are the open part, as they are for the Cauchy kernel above.

The Open Questions

1. Explicit forms. What are the explicit forms of the Poisson kernel and the Green's function in the oscillatory and nilpotent regimes?

2. The relation to Clifford analysis. How do these kernels relate to the corresponding kernels in Clifford analysis?

3. The Hardy spaces. What are the biquaternion analogues of the Hardy spaces $H^p$ and the Bergman spaces?

4. The reproducing kernel. Is there a reproducing kernel for the space of functions satisfying $\tilde{\nabla}\tilde{F} = 0$ on a domain?

These questions are open.

The General Principle

The higher special functions of a biquaternion variable fall into two classes.

Power-series functions. These are defined by a power series in $\tilde{Q}$, and the series can be summed in closed form in the two pure regimes (pure oscillatory and pure nilpotent). The Bessel functions, the orthogonal polynomials, and the hypergeometric function belong to this class. The general case (with both $Q_0 \neq 0$ and $\mathbf{Q} \neq 0$) does not reduce to a closed form in terms of the elementary functions, because the powers of $\tilde{Q}$ involve the non-commutative product $Q_0 \mathbf{Q}$.

Integral-defined functions. These are defined by an integral over a real or complex parameter, and the extension to biquaternions requires a notion of the integrand for biquaternion arguments. The gamma function and the zeta function belong to this class. The convergence and analyticity of the integral are not automatic, and the theory is largely open.

Functions from the analysis. These arise as kernels and fundamental solutions of the differential operators of the analysis. The Cauchy kernel, the Poisson kernel, and the Green's function belong to this class. Their construction is standard, but their explicit forms and their properties are open in the general biquaternion case.

The common thread is the complex norm $\theta$: the two regimes $\theta \neq 0$ and $\theta = 0$ determine the behavior of the powers of $\tilde{Q}$, and hence the behavior of every special function defined by a power series. The integral-defined functions depend on $\theta$ in a more complicated way, and their theory is less developed.

Open Questions

1. The general case for power-series functions. How do the Bessel functions, the orthogonal polynomials, and the hypergeometric function behave for a general biquaternion with both $Q_0 \neq 0$ and $\mathbf{Q} \neq 0$?

2. The gamma function. Can the integral defining $\Gamma(\tilde{Q})$ be made to converge for a general biquaternion, and does the functional equation hold?

3. The zeta function. Can the biquaternion zeta function be defined, and what are its properties? Is there an analogue of the Riemann hypothesis?

4. The special functions from the analysis. What are the explicit forms of the Poisson kernel, the Green's function, and the reproducing kernel for the spaces of functions satisfying $\tilde{\nabla}\tilde{F} = 0$ or $\Box\tilde{F} = 0$?

5. The relation to Clifford analysis. How do the biquaternion special functions relate to the corresponding functions in Clifford analysis?

6. The extension to several variables. Can the special functions be extended to functions of several biquaternion variables, and what replaces the two-regime structure?

Summary

The higher special functions of a biquaternion variable fall into three classes.

Power-series functions. The Bessel functions, the orthogonal polynomials, and the hypergeometric function are defined by power series, and the series can be summed in closed form in the two pure regimes. In the pure oscillatory regime, the functions are the ordinary functions evaluated at the complex argument $i\theta$ with $i$ replaced by the axis $\hat{n}$ of the component (for the Bessel function this is the modified function $I_n(\theta)$ times the parity factor $\hat{n}^n$). In the pure nilpotent regime, the series truncate to polynomials in $\mathbf{Q}$ of degree at most one. The general case is open.

Integral-defined functions. The gamma function and the zeta function are defined by integrals or series that require a notion of the biquaternion power for real arguments. The convergence and analyticity are established only in special cases, and the functional equations are open.

Functions from the analysis. The Cauchy kernel, the Poisson kernel, and the Green's function are the kernels and fundamental solutions of the differential operators of the analysis. Their construction is standard, but their explicit forms and properties in the general biquaternion case are open.

The common thread is the complex norm $\theta$: the two regimes $\theta \neq 0$ and $\theta = 0$ determine the behavior of the powers of $\tilde{Q}$, and hence the behavior of every special function defined by a power series. The integral-defined functions depend on $\theta$ in a more complicated way, and their theory is less developed.

The biquaternion special functions are therefore a partially developed subject: the power-series functions in the two pure regimes are well understood, while the general case and the integral-defined functions are open problems.

Physically the developed part is exactly the part the framework needs. The Bessel functions, the orthogonal polynomials and the hypergeometric function are the radial, angular and unifying modes of a separated wave equation, and the two pure regimes in which they are closed are the on-shell and lightlike cases the material sector supplies; the Cauchy, Poisson and Green's functions are the propagators and boundary kernels with which the field equations of Biquaternion Electromagnetism, Biquaternion Quantum Fields and Biquaternion Gravity and Cosmology are solved. The open part — the general mixed case and the integral-defined gamma and zeta functions — is the part that would be needed for an off-shell and a renormalised theory, and it is open because the biquaternion logarithm still has no extension to the light cone.

Summary of Notation

Further Reading

• William Rowan Hamilton, Lectures on Quaternions (Hodges and Smith, Dublin, 1853), for the original formulation.
• J. P. Ward, Quaternions and Cayley Numbers: Algebra and Applications (Kluwer, Dordrecht, 1997), Chapter 3, for the special functions of biquaternions.
• S. J. Sangwine, T. A. Ell, and N. Le Bihan, "Fundamental representations and algebraic properties of biquaternions or complexified quaternions", Advances in Applied Clifford Algebras 21 (2011) 607–636, for the polar forms.
• S. J. Sangwine, "Biquaternion (complexified quaternion) roots of $-1$", Advances in Applied Clifford Algebras 16 (2006) 63–68, for the roots of $-1$.
• S. J. Sangwine and D. Alfsmann, "Determination of the biquaternion divisors of zero, including the idempotents and nilpotents", Advances in Applied Clifford Algebras 20 (2010) 401–416, for the zero divisors and nilpotents.
• R. Fueter, "Die Funktionentheorie der Differentialgleichungen $\Delta u = 0$ und $\Delta\Delta u = 0$ mit vier reellen Variablen", Commentarii Mathematici Helvetici 7 (1934–35) 307–330, for the analysis of quaternion-valued functions of four real variables.
• F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis (Pitman, 1982), for the general Clifford analysis.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge, 2001), for the connection to Clifford algebras.