I know that we can define addition and multiplication (and other things related to the ring of formal power series like formal derivative) for formal power series over a ring but can we talk about fractional exponents? For instance $x^{\frac{3}{2}}$? Solving problems by generating functions lead me ask this question. Actually, I need to know if it's necessary to look at the problems with an analytic view (paying attention to and using things like the radius of convergence) or not. We have to show that fractional exponentiation is well-defined in the context of formal power series in order to use it.
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I don't see why not. Have you tried it? Did you encounter any difficulties? Radius of convergence is not an issue with formal power series. – Somos Aug 16 '21 at 15:30
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@Somos So you mean radius of convergence is defined for any formal power series? Let me explain more: For some power series we can make sure of the validity of calculations by replacing x with a point inside the disk of convergence of the Maclaurin series. For instance, you can say the power series of the Catalan sequence is equal to $\frac{1 - \sqrt{1 - 4x}}{2x}$ where $x$ is any point inside its disk of convergence so you don't need to look at it from an algebraic point of view. – Emad Aug 16 '21 at 16:35
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@Somos . You don't need to say that $\frac{1 - \sqrt{1 - 4x}}{2x}$ is indeed a power series and equal to $\sum_{n = 0}^{+\infty}c_{n}x^{n}$. And you don't need to explain why raising $1 - 4x$ which is a polynomial (and as a result a power series) to the power of $\frac{1}{2}$ is allowed. That's what I say. Can we interpret it algebraically? – Emad Aug 16 '21 at 16:36
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The Wikipedia article Puiseux series may help answer your question. – Somos Aug 16 '21 at 21:23