
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12566-2
10.1016/j.heliyon.2024.e36535
e36535
Research Article
A quadratic transformation for a special confluent Heun function
Ishkhanyan A.M. aishkhanyan@gmail.com

Institute for Physical Research, Ashtarak, 0204, Armenia
20 8 2024
30 8 2024
20 8 2024
10 16 e3653520 11 2023
25 6 2024
19 8 2024
© 2024 The Author
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
We report a functional identity involving a quadratic transformation of the argument for the confluent Heun function. This identity is the first of its kind to be discovered for the confluent Heun function, and it has the potential to be useful in a variety of applications.

Keywords

Confluent Heun function
Quadratic transformation
Hypergeometric function
MSC

33E30
33E10
34B30
==== Body
pmcThe confluent Heun function, denoted as HeunC(q,α,γ,δ,ε,z), is the regular solution of the confluent Heun equation:(1) d2udz2+(γz+δz−1+ε)dudz+αz−qz(z−1)u=0,

normalized to 1 at the origin [1,2]. As a special function of the new generation, it extends the domain of many classical special functions and has extensive applications in contemporary physics and mathematics, ranging from classical to quantum physics, general relativity, and cosmology [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]]. However, despite considerable efforts, this function remains relatively unexplored compared to its hypergeometric counterparts. While many functional identities involving quadratic or cubic transformations of the argument are known for the ordinary hypergeometric functions [18], no such identities have been discovered for the confluent Heun functions. This is the topic we discuss in this brief note, and we report an identity involving a quadratic transformation of the argument.

Our result is that for 0<z<1 or any z with Im(z)≠0, the following relationship holds:(2) zeεz2HeunC(q,ε,2,0,ε,z)=C1HeunC(q−ε4−ε264,−ε264,12,0,0,(1−2z)2)+C2(1−2z)HeunC(q−ε4−ε264,−ε264,32,0,0,(1−2z)2),

where(3) C1=eε/42HeunC(q,ε,2,0,ε,12)

and(4) C2=−4+ε4C1−eε/44HeunCPrime(q,ε,2,0,ε,12).

here, HeunCPrime represents the derivative of the confluent Heun function. This relation can be verified directly by substituting u(z)=e−εz/2v(y)/z and y=(1−2z)2 into equation (1).

A side result emerges when the presented relation is simplified into simpler special functions, occurring in two specific cases, apart from polynomial reductions.

First, the confluent Heun function on the left-hand side of equation (2) is reduced to the Kummer confluent hypergeometric function when q=α, that is q=ε. Furthermore, given δ=0, the resulting confluent hypergeometric function is further simplified to an elementary function, and the left-hand side of the equation becomes 2sinh(εz/2). Since the right-hand side of equation (2) produces the same function, we do not obtain any new results in this case.

Another reduction of the confluent Heun function on the left-hand side of equation (2) is achieved when α=ε=0. The confluent Heun function is then reduced to the Gauss ordinary hypergeometric function. Similar reductions occur with all terms on the right-hand side of equation (2), except for the term involving the HeunCPrime function. As a result, we arrive at the following representation of a special HeunCPrime function with the argument 1/2:(5) HeunCPrime(q,0,2,0,0,12)=2πΓ(1−a2)Γ(a+12)−sin(πa)Γ(a−12)Γ(−a2)2π3/2,

where a=12(1±1+4q). (6)

Here q can be either real or complex. We have tested the derived relations through extensive simulations using Mathematica's Heun functions [19].

We conclude by noting that confluent Heun functions with parameters involved in equations (2), (3), (4), (5) appear in several physical problems. We, therefore, envisage many applications of the presented relations. For example, the function on the left-hand side of equation (2) with the following specializations:(7) ε=±32mEℏ2,q=ε−2mV0ℏ2,z=1+ix2,

is the solution of the stationary one-dimensional Schrödinger equation for a particle of mass m and energy E in the Lorentzian potential well:(8) V(x)=−V01+x2.

An accompanying observation here is that the Schrödinger equation for this potential is invariant with respect to the reflection x→−x. Hence, one may apply the approach outlined in Ref. [20], Sect. 2.4, to construct a solution depending on the quadratic argument x2. It turns out that this solution obeys a confluent Heun equation, and one then recovers relation (2).

Funding

This work was supported by the 10.13039/501100007029 Armenian State Science Committee grant no. 21AG-1C064 .

Data availability statement

No data was used for the research described in the article.

CRediT authorship contribution statement

A.M. Ishkhanyan: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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