van der Toorn, R. Tandem Recurrence Relations for Coefficients of Logarithmic Frobenius Series Solutions about Regular Singular Points. Axioms2023, 12, 32.
van der Toorn, R. Tandem Recurrence Relations for Coefficients of Logarithmic Frobenius Series Solutions about Regular Singular Points. Axioms 2023, 12, 32.
van der Toorn, R. Tandem Recurrence Relations for Coefficients of Logarithmic Frobenius Series Solutions about Regular Singular Points. Axioms2023, 12, 32.
van der Toorn, R. Tandem Recurrence Relations for Coefficients of Logarithmic Frobenius Series Solutions about Regular Singular Points. Axioms 2023, 12, 32.
Abstract
We enhance Frobenius' method for solving linear ordinary differential equations about regular singular points. Key to Frobenius' approach is the exploration of the derivative with respect to a single parameter; this parameter is introduced through the powers of generalized power series. Extending this approach, we discover that tandem recurrence relations can be derived. These relations render coefficients for series occurring in logarithmic solutions. The method applies to the, practically important, exceptional cases in which the roots of the indicial equation are equal, or differ by a non-zero integer. We demonstrate the method on Bessel's equation and derive previously unknown tandem recurrence relations for coefficients of solutions of the second kind, for Bessel equations of all integer and half-integer order.
Keywords
ordinary differential equations; Frobenius method; tandem recurrence relations; Bessel's equation
Subject
MATHEMATICS & COMPUTER SCIENCE, General Mathematics
Copyright:
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