# Tandem Recurrence Relations for Coefficients of Logarithmic Frobenius Series Solutions about Regular Singular Points

## Abstract

**:**

## 1. Introduction

#### 1.1. Subject, Problem, Main Goal and Results of This Manuscript

#### 1.1.1. Main Subject of This Manuscript: Frobenius’ Method

#### 1.1.2. Current Status of Frobenius’ $d/dr$ Method and Problem Statement

#### 1.1.3. Aim of This Manuscript

#### 1.1.4. Main Result

#### 1.1.5. Demonstration

#### 1.2. Frobenius’ Method

#### 1.2.1. Purpose of the Method; Regular Singular Points

#### 1.2.2. Frobenius’ Ansatz and the Merits of His Approach

#### 1.2.3. On the Results of Fuchs

#### 1.2.4. On What Frobenius Added to the Theory of Fuchs. Enhanced $d/dr$-Method

#### 1.3. Tandem Recurrence Relations Emerging from the $d/dr$ Method

#### 1.3.1. Relevance of Frobenius’ $d/dr$-Method and Our Enhancement of It

#### 1.3.2. Current Status of $d/dr$-Method and Novelty of Our Enhancement

#### 1.3.3. Merit of Our Tandem Approach

#### 1.4. To Normalize or Not to Normalize the Coefficient of the Highest Order Derivative

#### 1.4.1. Original Goal of the Theory

#### 1.4.2. The Normalization Convention in Textbooks

#### 1.4.3. Why Normalization May Induce Complication

#### 1.4.4. Why Polynomial Coefficients Can Bring Computational Efficiency

#### 1.5. Outline

#### 1.5.1. Exposition of the Theory

#### 1.5.2. Demonstration: Bessel’s Equation

#### 1.5.3. Summary, Reflection and History

## 2. Analysis and Enhancement of Frobenius’ Method

#### 2.1. Frobenius’ Expansion of the Image of the Differential Operator, First Solution of the Differential Equation and Second Solution When the Roots of the Indicial Equation Are Unequal and Do Not Differ by an Integer

#### 2.1.1. General Expansion of the Image of the Differential Operator

#### 2.1.2. Solution Associated with the Largest Root of the Indicial Equation

#### 2.1.3. Second Linearly Independent Solution? Emergence of the So-Called Exceptional Cases

#### 2.2. Conception of the $d/dr$ Method, General Relations Central to the Method and Second Linearly Independent Solution in Case the Roots of the Indicial Equation Are Equal

#### 2.2.1. Inspiration from the Case in Which the Roots of the Indicial Equation Are Equal

#### 2.2.2. Frobenius’ Class of Functions $\tilde{y}(x,r)$ and General Relations Central to the $d/dr$ Method

#### 2.2.3. First Application of the $d/dr$ Method: Solutions in Case the Roots ${r}_{1}$ and ${r}_{2}$ of the Indicial Equation Are Equal

#### 2.2.4. The Derivative of the First Recurrence Relation with Respect to Frobenius’ Parameter r

#### 2.2.5. Tandem Recurrence Relations for Generalized Power Series Coefficients in Case ${r}_{1}={r}_{2}$

#### 2.2.6. The Novelty of, and Enhancement Established by, the Tandem Technique

#### 2.3. Solutions Associated with ${r}_{2}$ in Case ${r}_{1}-{r}_{2}=N$

#### 2.3.1. Structure of Indicial Polynomial

#### 2.3.2. With $r={r}_{2}$ and the Free Coefficient ${a}_{N}$, a Solution Linearly Dependent on $y(x,{r}_{1})$ Is Associated

**Remark**

**1.**

**Remark**

**2.**

#### 2.3.3. Second Linearly Independent Solution in Case ${r}_{1}-{r}_{2}=N$

#### 2.3.4. Tandem Recurrence Relations for the Coefficients ${c}_{n}$

#### 2.3.5. Why ${c}_{N}$ Remains a Free Parameter and the Meaning of this

#### 2.3.6. How to Obtain ${a}_{N}\left({r}_{2}\right)$, the Coefficient of the Logarithmic Factor

#### 2.3.7. Intermediate Summary of the Method for Case ${r}_{1}-{r}_{2}=N$

#### 2.3.8. Possibility of Solutions Associated with ${r}_{2}$ without a Logarithmic Term

“(…) no modification in the definition of ${J}_{\nu}\left(z\right)$ is necessary when $\nu $ (is half-integer); the real peculiarity of the solution in this case is that the negative root of the indicial equation gives rise to a series containing two arbitrary constants, i.e., to the general solution of the differential equation.”

#### 2.3.9. Conclusions

## 3. Application: Series Solutions for Bessel’s Equation

#### 3.1. Coefficients and Indicial Polynomial

#### 3.2. Minimal Value of n for the Recurrence Relations to Be Form Invariant

#### 3.3. Recurrence Relations for $n=1$

#### 3.4. Recurrence Relations for $y(x,\nu )$ and $2\le n$

#### 3.5. Results Relevant to Bessel Functions of the Second Kind

#### 3.5.1. Distinct Roots of the Indicial Equation, Not Differing by an Integer

#### 3.5.2. Two Equal Roots: Bessel Equation of Order Zero

#### 3.5.3. Bessel Equations of Integer or Half-Integer Order Larger than Zero

## 4. Discussion

#### 4.1. Synopsis and Goal of Further Discussion

#### 4.2. Historical Origin and Background

- The standardization of the differential equation by normalizing the pre-factor of ${y}^{\u2033}$ to be ${x}^{2}$. In addition, see Figure 2.
- The Ansatz (5), inspired by the earlier results of Fuchs, to look for solutions in the form of generalized power series, while this generalization introduces no more than only a single parameter r. This parameter is used to shift the powers of the variable x in the power series, all by the same amount.
- Frobenius’ discovery that, at least for a first series solution, the coefficients ${a}_{n}$ of the series obey a recurrence relation that can be derived quite simply (“einfach”, see Figure 3).
- The idea that then all coefficients ${a}_{n}\left(r\right)$ of the series may be conceived as functions of r, too.
- As an attempt to construct solutions associated with a second root, ${r}_{2}$, of the indicial equation, in cases of roots ${r}_{1}$ and ${r}_{2}$ being equal or differing by an integer, Frobenius explored the consequences of choosing the leading coefficient ${a}_{0}\left(r\right)$ such that a division by zero in recurrence relations for the ${a}_{n}$ would be replaced by a limit process, ${lim}_{r\to {r}_{2}}$, so as to obtain finite values for all coefficients ${a}_{n}$. That is, in our phrasing, Frobenius explored the option to have$${a}_{0}\left(r\right)={c}_{0}\phantom{\rule{0.166667em}{0ex}}(r-{r}_{2})\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}$$
- Frobenius’ idea that by means of differentiation with respect to r, new solutions of the differential equation can be obtained.

#### 4.3. The Role and History of the Factor $r-{r}_{2}$ in the Literature

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The title and opening sentence of Frobenius’ original paper on series solutions of linear differential equations [6]. Note that, as in the works of Fuchs [11,12], differential equations of n-th order were addressed, in full generality. To avoid prolixity and to allow focus instead, in our manuscript here, we restrict our coverage to second order equations. Frobenius introduced the name $P\left(y\right)$ for the left hand side of the equation shown here. This corresponds to our linear operator L, i.e., to our expression (2).

**Figure 2.**Frobenius’ Ansatz, our Equation (5), as published in 1873 [6], at the top of the third page of his paper. This fragment also contains the choice to normalize the coefficient of the highest order derivative, i.e., to set “$p\left(x\right)=1$”, Zur Vereinfachung der Beweise (“To simplify the proof”).

**Figure 3.**Closing paragraph of the opening section of Frobenius’ seminal paper on series solutions [6]. While paying due respect to the earlier works of Weierstrass and Fuchs, Frobenius explained that he believed his own approach offered a worthwhile alternative, because of its simplicity (“ …einfach …”), ease (“ …mit Leichtigkeit …”) and directness (“ …direct …”). Three decades later, Forsyth ([1], Chap. VI) would indeed recommend Frobenius’ process, because of these merits.

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**MDPI and ACS Style**

van der Toorn, R.
Tandem Recurrence Relations for Coefficients of Logarithmic Frobenius Series Solutions about Regular Singular Points. *Axioms* **2023**, *12*, 32.
https://doi.org/10.3390/axioms12010032

**AMA Style**

van der Toorn R.
Tandem Recurrence Relations for Coefficients of Logarithmic Frobenius Series Solutions about Regular Singular Points. *Axioms*. 2023; 12(1):32.
https://doi.org/10.3390/axioms12010032

**Chicago/Turabian Style**

van der Toorn, Ramses.
2023. "Tandem Recurrence Relations for Coefficients of Logarithmic Frobenius Series Solutions about Regular Singular Points" *Axioms* 12, no. 1: 32.
https://doi.org/10.3390/axioms12010032