Bicomplex Bessel Function

• The bicomplex Bessel function is a bicomplex extension of the classical Bessel function, defined via a power series and idempotent decomposition that represents it as a pair of complex Bessel functions.
• It employs classical analytic tools such as gamma functions, recurrence relations, integral representations, and differential equations, extended to the bicomplex hypergeometric framework.
• The function underpins applications in advanced PDE theory, bicomplex transforms, and coherent state quantization, illustrating its significance in mathematical physics and signal processing.

Searching arXiv for the cited bicomplex Bessel and related bicomplex hypergeometric papers to ground the article. The bicomplex Bessel function is a bicomplex-valued extension of the classical Bessel function of the first kind in which both the order and the argument are allowed to lie in the bicomplex algebra $\mathbb{BC}$. In the formulation introduced in "On the Bessel function and $n$-dimensional Hankel transform with Bicomplex arguments and coherent states" (Bera et al., 22 Jul 2025), it retains the classical analytic apparatus—power-series definition, recurrence relations, integral representations, differential equations, asymptotics, and transform theory—while being governed by the idempotent decomposition of bicomplex numbers. A central structural fact is that the bicomplex Bessel function is represented exactly by a pair of ordinary complex Bessel functions encoded in the idempotent basis, so the theory is simultaneously a genuine bicomplex function theory and a componentwise lifting of classical Bessel analysis (Bera et al., 22 Jul 2025).

1. Algebraic framework

The ambient ring is

$$\mathbb{BC}=\{\lambda_1+j\lambda_2=a+ib+jc+kd:\ a,b,c,d\in\mathbb{R},\ \lambda_1,\lambda_2\in\mathbb{C}(i)\},$$

with

$i^2=j^2=-1,\qquad k=ij=ji,\qquad k^2=1.$

Its decisive structural feature is the idempotent basis

$e_1=\frac{1+k}{2},\qquad e_2=\frac{1-k}{2},$

satisfying

$$e_1^2=e_1,\quad e_2^2=e_2,\quad e_1e_2=0,\quad e_1+e_2=1.$$

Every bicomplex number has a unique decomposition

$Z=z_1e_1+z_2e_2,$

with $z_1,z_2\in\mathbb{C}(i)$, and the associated projections are

$$\mathcal{S}_1(Z)=z_1=\lambda_1-i\lambda_2,\qquad \mathcal{S}_2(Z)=z_2=\lambda_1+i\lambda_2.$$

Convergence and growth are measured with the hyperbolic norm

$|Z|_h=|z_1|e_1+|z_2|e_2,$

which satisfies $n$0. The null cone of zero divisors is

$n$1

Bicomplex holomorphicity is defined by bicomplex differentiability, equivalently by holomorphicity of the scalar components together with the bicomplex Cauchy–Riemann equations

$n$2

These identities are the technical substrate for the entire theory of bicomplex special functions (Bera et al., 22 Jul 2025).

A second prerequisite is the bicomplex gamma function, defined by an Euler product and decomposing as

$n$3

This decomposition is what allows classical coefficient formulas to be transferred to the bicomplex setting without altering their scalar content (Bera et al., 22 Jul 2025).

2. Hypergeometric provenance

The bicomplex Bessel function sits naturally inside the theory of bicomplex generalized hypergeometric functions developed in "Bicomplex generalized hypergeometric functions and their applications" (Bera et al., 2023). For bicomplex parameters $n$4 and bicomplex variable $n$5, that paper defines

$n$6

and proves the idempotent representation

$n$7

The convergence theorem in that framework states that if $n$8, the series is absolutely hyperbolically convergent for all $n$9, and the analyticity theorem states that $$\mathbb{BC}=\{\lambda_1+j\lambda_2=a+ib+jc+kd:\ a,b,c,d\in\mathbb{R},\ \lambda_1,\lambda_2\in\mathbb{C}(i)\},$$0 is $$\mathbb{BC}=\{\lambda_1+j\lambda_2=a+ib+jc+kd:\ a,b,c,d\in\mathbb{R},\ \lambda_1,\lambda_2\in\mathbb{C}(i)\},$$1-holomorphic in $$\mathbb{BC}=\{\lambda_1+j\lambda_2=a+ib+jc+kd:\ a,b,c,d\in\mathbb{R},\ \lambda_1,\lambda_2\in\mathbb{C}(i)\},$$2 and in the parameters except at denominator singularities (Bera et al., 2023). The special case $$\mathbb{BC}=\{\lambda_1+j\lambda_2=a+ib+jc+kd:\ a,b,c,d\in\mathbb{R},\ \lambda_1,\lambda_2\in\mathbb{C}(i)\},$$3, which is the classical hypergeometric form underlying Bessel functions, is therefore available in bicomplex analysis with global convergence.

The later Bessel paper uses precisely this mechanism. It rewrites the bicomplex Bessel function in the form

$$\mathbb{BC}=\{\lambda_1+j\lambda_2=a+ib+jc+kd:\ a,b,c,d\in\mathbb{R},\ \lambda_1,\lambda_2\in\mathbb{C}(i)\},$$4

and then derives the bicomplex Bessel differential equation by specializing the general bicomplex hypergeometric differential equation for $$\mathbb{BC}=\{\lambda_1+j\lambda_2=a+ib+jc+kd:\ a,b,c,d\in\mathbb{R},\ \lambda_1,\lambda_2\in\mathbb{C}(i)\},$$5 (Bera et al., 22 Jul 2025). In this sense, the bicomplex Bessel function is not isolated; it is a distinguished instance of a broader bicomplex hypergeometric calculus.

3. Definition and componentwise structure

If

$$\mathbb{BC}=\{\lambda_1+j\lambda_2=a+ib+jc+kd:\ a,b,c,d\in\mathbb{R},\ \lambda_1,\lambda_2\in\mathbb{C}(i)\},$$6

the bicomplex Bessel function of order $$\mathbb{BC}=\{\lambda_1+j\lambda_2=a+ib+jc+kd:\ a,b,c,d\in\mathbb{R},\ \lambda_1,\lambda_2\in\mathbb{C}(i)\},$$7 is defined by the power series

$$\mathbb{BC}=\{\lambda_1+j\lambda_2=a+ib+jc+kd:\ a,b,c,d\in\mathbb{R},\ \lambda_1,\lambda_2\in\mathbb{C}(i)\},$$8

This is a direct bicomplex analogue of the classical series

$$\mathbb{BC}=\{\lambda_1+j\lambda_2=a+ib+jc+kd:\ a,b,c,d\in\mathbb{R},\ \lambda_1,\lambda_2\in\mathbb{C}(i)\},$$9

The defining theorem of the subject is the idempotent decomposition

$i^2=j^2=-1,\qquad k=ij=ji,\qquad k^2=1.$0

Accordingly,

$i^2=j^2=-1,\qquad k=ij=ji,\qquad k^2=1.$1

The normalized series $i^2=j^2=-1,\qquad k=ij=ji,\qquad k^2=1.$2 converges absolutely in the hyperbolic sense for all $i^2=j^2=-1,\qquad k=ij=ji,\qquad k^2=1.$3, with infinite hyperbolic radius of convergence (Bera et al., 22 Jul 2025).

This decomposition has strong conceptual consequences. The bicomplex Bessel function is not an irreducibly new scalar special function in the complex sense; it is exactly the pair $i^2=j^2=-1,\qquad k=ij=ji,\qquad k^2=1.$4 encoded by $i^2=j^2=-1,\qquad k=ij=ji,\qquad k^2=1.$5 and $i^2=j^2=-1,\qquad k=ij=ji,\qquad k^2=1.$6. All bicomplex identities are therefore bicomplex recombinations of two classical identities, but they are organized within a single algebra with its own norm, null cone, holomorphicity theory, and partial order (Bera et al., 22 Jul 2025).

The paper also proves a symmetry analogous to $i^2=j^2=-1,\qquad k=ij=ji,\qquad k^2=1.$7 when the idempotent components of the order are non-positive integers, and derives a symmetry with respect to bicomplex index shifts $i^2=j^2=-1,\qquad k=ij=ji,\qquad k^2=1.$8 and $i^2=j^2=-1,\qquad k=ij=ji,\qquad k^2=1.$9. These are direct lifts of componentwise classical order symmetries (Bera et al., 22 Jul 2025).

4. Identities, integral representations, and differential equation

The bicomplex Bessel function satisfies order recurrences that mirror the classical ones. Among the formulas established are

$e_1=\frac{1+k}{2},\qquad e_2=\frac{1-k}{2},$0

and

$e_1=\frac{1+k}{2},\qquad e_2=\frac{1-k}{2},$1

Because of the idempotent form, these are exactly the recurrences for $e_1=\frac{1+k}{2},\qquad e_2=\frac{1-k}{2},$2 and $e_1=\frac{1+k}{2},\qquad e_2=\frac{1-k}{2},$3 written in bicomplex notation (Bera et al., 22 Jul 2025).

Several integral representations are derived under explicit parameter restrictions. They include a beta-function-type representation, a trigonometric representation valid under the condition $e_1=\frac{1+k}{2},\qquad e_2=\frac{1-k}{2},$4, a double-beta formula for shifted order, and a Laplace-type representation obtained from Legendre’s duplication formula and the classical Laplace integral for $e_1=\frac{1+k}{2},\qquad e_2=\frac{1-k}{2},$5. The generating function for integer order is

$e_1=\frac{1+k}{2},\qquad e_2=\frac{1-k}{2},$6

together with a contour-integral representation obtained by bicomplex Laurent theory (Bera et al., 22 Jul 2025).

Differentiation with respect to the bicomplex variable yields the neighboring-order relations

$e_1=\frac{1+k}{2},\qquad e_2=\frac{1-k}{2},$7

and

$e_1=\frac{1+k}{2},\qquad e_2=\frac{1-k}{2},$8

Using the hypergeometric representation and the bicomplex $e_1=\frac{1+k}{2},\qquad e_2=\frac{1-k}{2},$9 differential equation, the paper proves that $$e_1^2=e_1,\quad e_2^2=e_2,\quad e_1e_2=0,\quad e_1+e_2=1.$$0 satisfies

$$e_1^2=e_1,\quad e_2^2=e_2,\quad e_1e_2=0,\quad e_1+e_2=1.$$1

In idempotent components this reduces to the two ordinary Bessel equations

$$e_1^2=e_1,\quad e_2^2=e_2,\quad e_1e_2=0,\quad e_1+e_2=1.$$2

$$e_1^2=e_1,\quad e_2^2=e_2,\quad e_1e_2=0,\quad e_1+e_2=1.$$3

The bicomplex differential equation is therefore a direct generalization rather than a formally unrelated equation (Bera et al., 22 Jul 2025).

5. Holomorphicity, asymptotics, and analytic behavior

Two holomorphicity questions are treated separately: dependence on the order and dependence on the argument. For fixed $$e_1^2=e_1,\quad e_2^2=e_2,\quad e_1e_2=0,\quad e_1+e_2=1.$$4, the function $$e_1^2=e_1,\quad e_2^2=e_2,\quad e_1e_2=0,\quad e_1+e_2=1.$$5 is bicomplex holomorphic in $$e_1^2=e_1,\quad e_2^2=e_2,\quad e_1e_2=0,\quad e_1+e_2=1.$$6. This is obtained by rewriting

$$e_1^2=e_1,\quad e_2^2=e_2,\quad e_1e_2=0,\quad e_1+e_2=1.$$7

in the $$e_1^2=e_1,\quad e_2^2=e_2,\quad e_1e_2=0,\quad e_1+e_2=1.$$8-basis and then invoking the classical holomorphicity of $$e_1^2=e_1,\quad e_2^2=e_2,\quad e_1e_2=0,\quad e_1+e_2=1.$$9 in the order together with the bicomplex Cauchy–Riemann equations (Bera et al., 22 Jul 2025).

Holomorphicity in the argument is proved under discrete constraints on the order. If

$Z=z_1e_1+z_2e_2,$0

with

$Z=z_1e_1+z_2e_2,$1

then $Z=z_1e_1+z_2e_2,$2 and $Z=z_1e_1+z_2e_2,$3, so $Z=z_1e_1+z_2e_2,$4 and $Z=z_1e_1+z_2e_2,$5 are entire in $Z=z_1e_1+z_2e_2,$6. Under these conditions, $Z=z_1e_1+z_2e_2,$7 is holomorphic in $Z=z_1e_1+z_2e_2,$8 (Bera et al., 22 Jul 2025).

The asymptotic theory is formulated in hyperbolic terms. A bicomplex function $Z=z_1e_1+z_2e_2,$9 has an asymptotic expansion

$z_1,z_2\in\mathbb{C}(i)$0

if for each fixed $z_1,z_2\in\mathbb{C}(i)$1,

$z_1,z_2\in\mathbb{C}(i)$2

Starting from the hypergeometric representation and using a relation between $z_1,z_2\in\mathbb{C}(i)$3 and $z_1,z_2\in\mathbb{C}(i)$4, the authors derive a large-$z_1,z_2\in\mathbb{C}(i)$5 expansion by a stationary-phase-like argument. Under the condition $z_1,z_2\in\mathbb{C}(i)$6, the leading factor is

$z_1,z_2\in\mathbb{C}(i)$7

multiplied by bicomplex gamma-function coefficients and a series in inverse powers of $z_1,z_2\in\mathbb{C}(i)$8 involving the Pochhammer symbol (Bera et al., 22 Jul 2025). This extends a classical large-argument asymptotic formula to the bicomplex setting.

6. Hankel transform, partial differential equations, and coherent states

The bicomplex Bessel function is the kernel of the $z_1,z_2\in\mathbb{C}(i)$9-dimensional bicomplex Hankel transform. For $$\mathcal{S}_1(Z)=z_1=\lambda_1-i\lambda_2,\qquad \mathcal{S}_2(Z)=z_2=\lambda_1+i\lambda_2.$$0 in a weighted test-function space and $$\mathcal{S}_1(Z)=z_1=\lambda_1-i\lambda_2,\qquad \mathcal{S}_2(Z)=z_2=\lambda_1+i\lambda_2.$$1 in a bicomplex domain $$\mathcal{S}_1(Z)=z_1=\lambda_1-i\lambda_2,\qquad \mathcal{S}_2(Z)=z_2=\lambda_1+i\lambda_2.$$2, the transform is defined by

$$\mathcal{S}_1(Z)=z_1=\lambda_1-i\lambda_2,\qquad \mathcal{S}_2(Z)=z_2=\lambda_1+i\lambda_2.$$3

The relevant function spaces are $$\mathcal{S}_1(Z)=z_1=\lambda_1-i\lambda_2,\qquad \mathcal{S}_2(Z)=z_2=\lambda_1+i\lambda_2.$$4, a smooth bicomplex test space defined by hyperbolic seminorm bounds in the $$\mathcal{S}_1(Z)=z_1=\lambda_1-i\lambda_2,\qquad \mathcal{S}_2(Z)=z_2=\lambda_1+i\lambda_2.$$5-variables, and $$\mathcal{S}_1(Z)=z_1=\lambda_1-i\lambda_2,\qquad \mathcal{S}_2(Z)=z_2=\lambda_1+i\lambda_2.$$6, a smooth bicomplex space controlled by exponential-type growth estimates in $$\mathcal{S}_1(Z)=z_1=\lambda_1-i\lambda_2,\qquad \mathcal{S}_2(Z)=z_2=\lambda_1+i\lambda_2.$$7. The transform is shown to be a continuous linear map

$$\mathcal{S}_1(Z)=z_1=\lambda_1-i\lambda_2,\qquad \mathcal{S}_2(Z)=z_2=\lambda_1+i\lambda_2.$$8

and, with the inversion formula on real arguments, an isomorphism of bicomplex topological vector spaces between these two spaces (Bera et al., 22 Jul 2025).

The transform intertwines bicomplex differential operators $$\mathcal{S}_1(Z)=z_1=\lambda_1-i\lambda_2,\qquad \mathcal{S}_2(Z)=z_2=\lambda_1+i\lambda_2.$$9 and $|Z|_h=|z_1|e_1+|z_2|e_2,$0 with multiplication by $|Z|_h=|z_1|e_1+|z_2|e_2,$1 and $|Z|_h=|z_1|e_1+|z_2|e_2,$2. In particular,

$|Z|_h=|z_1|e_1+|z_2|e_2,$3

This diagonalization property is used to solve bicomplex generalized wave and heat equations. For $|Z|_h=|z_1|e_1+|z_2|e_2,$4, the wave equation becomes

$|Z|_h=|z_1|e_1+|z_2|e_2,$5

and the heat equation becomes

$|Z|_h=|z_1|e_1+|z_2|e_2,$6

For $|Z|_h=|z_1|e_1+|z_2|e_2,$7, these reduce respectively to the classical one-dimensional wave and heat equations (Bera et al., 22 Jul 2025).

The same paper constructs bicomplex generalized coherent states whose normalization and overlap are governed by $|Z|_h=|z_1|e_1+|z_2|e_2,$8. In a bicomplex Fock basis $|Z|_h=|z_1|e_1+|z_2|e_2,$9, the coherent states are

$n$00

They satisfy

$n$01

and their overlap is

$n$02

The annihilation operator acts by

$n$03

so the states are coherent in the standard eigenstate sense. The paper further proves continuity in $n$04 and a resolution of the identity using a positive bicomplex weight whose idempotent components are expressed through Meijer $n$05-functions (Bera et al., 22 Jul 2025).

Conceptually, the later sections clarify the role of the bicomplex Bessel function in analysis and mathematical physics. Via idempotent decomposition, the bicomplex Hankel transform is a pair of classical Hankel transforms, and the transformed PDEs correspond to two classical PDEs coupled only through the bicomplex algebra. The authors mention applications in mathematical physics, bicomplex quantum mechanics, signal processing, PDE theory, and coherent state quantization. A plausible implication is that the bicomplex framework is most useful when two related complex processes or two spectral sectors are to be treated simultaneously within one algebraic and analytic formalism (Bera et al., 22 Jul 2025, Bera et al., 2023).