@nm Here is an example: 

https://www.maplesoft.com/support/help/Maple/view.aspx?path=hypergeom

By implementing the substitution at the final stage—specifically, replacing BesselI(0, s) with (1/s - BesselI(1, s) * BesselK(0, s))/BesselK(1, s)—all the expressions can be elegantly streamlined. This simplification extends seamlessly to the subsequent calculations, which I have omitted for brevity. I sincerely appreciate your valuable input and suggestions.

@Thomas Richard Enclosed is a concise script elucidating the issue at hand. Essentially, the solution to the linear system of equations exhibits a straightforward form, as previously achieved with a Maple license (likely Maple 20) at my former workplace. Having recently transitioned to a new workplace, I am currently testing the trial version of Maple 23, installed just yesterday. Upon running the same script, I am confronted with convoluted expressions. The attached script provides additional details. Any hints or insights would be greatly appreciated. Alternatively, reporting this issue to the development team might serve to bring their attention to the problem.

question2.mw

A research-oriented script addressing a persistent fluid mechanics problem ran smoothly on my prior Maple 20 without the need for 'wronskian' to simplify expressions. However, upon attempting the latest trial release, I encountered messy expressions. This led me to contemplate whether trial versions intentionally lack certain packages, potentially explaining their reduced functionality...

@Thomas Richard Thank you for your response. It appears that Maple 20, or thereabouts, performs more effectively than the trial version of Maple 23. I am currently facing simplification challenges that were not present in my experience with Maple 20.

@dharr Thanks for the info. I attach an example where this can be seen. I have experience with this issue so I also try to choose a much larger order to make sure that the series is evaluated at the desired order. I can always remove the ones that I do not need. It is worth mentioning that by executing the script twice one gets the desired result. However, it is not always a good idea to execute scripts twice before obtaining the results we are looking for.

question.mw

@sursumCorda OK. Thanks. Sometimes the series expansion stops much earlier but this not the problem. One can simply specify a larger number n and then extract the terms with negative exponent. 

@vv Thanks for the clarification. It turned out that (1,0) in power is simply the derivative of the regularized hypergeom with respect to the first argument, i.e., a. I am now able to obtain the same value,