A few years ago, some colleagues from institution A (where I also work) developed a certain technique. They tested it experimentally, it worked and gave good results. They wrote it up and got it published, but it lacked a rigorous mathematical analysis of errors, etc.
I took an interest in this, and thought of an improved technique that in theory should give excellent results, significantly better than the original one. Testing it experimentally would require a lot of effort and time, so I decided to perform some mathematical and computational analysis and simulation to make sure my hunch about improved performance was correct. The analysis techniques I used by themselves are not novel, but still, they thoroughly considered every source of error/bias in this technique, resulting in dozens of pages of algebra, CAS (Computer Algebra System) scripts and simulations in specialized software. All of this confirmed that performance should indeed improve. I loaded all these documents into a shared folder, accessible by my colleagues.
Then, my colleagues decided to proceed with building the experimental apparatus. Soon afterwards, I left temporarily for institution B, so I lost close contact with my colleagues of institution A (although we still talk from time to time, just much less than before). However, I will return to institution A in the future. I learned that they did try building the apparatus using my improved technique, but were unable to solve some problems that showed up along the way, so they don't have any results based on this technique yet.
Today I stumbled across a recently published paper by these colleagues -- to be clear, they didn't alert me to it or anything, I found out about this 100% on my own. Skimming over this paper, my understanding is that they basically took their old technique, made a few cursory improvements to some weak points, and most importantly, included a thorough analysis of this technique which, despite having only skimmed over the paper, I am fairly certain is a direct adaptation of the work I did. This was published in a high impact factor journal and I am fairly certain it wouldn't have been accepted without this thorough analysis. Moreover, all the documentation that I left for them, although specific to the improved technique, would be easily adapted to this old technique. My colleagues never asked me to coauthor the paper and didn't thank or otherwise acknowledge my work. Also, I never even learned this paper was being written or got published, even though I spoke to one of the coauthors (about unrelated subjects) a few times in the timeframe this paper was being written -- while I may be wrong, I get the feeling they were trying to hide its existence from me.
To make a long story short, I feel like this paper getting published owes a lot to the use of my analysis techniques applied to their old technique. As such, I feel they should have contacted me and offered me authorship. Am I wrong in thinking this?
For those who agree that there was at least some degree of fault here: what are some possible ways of addressing this with my colleagues? I do not intend to make a mess out of this (by e.g. contacting the editor of the journal and demanding credit), since we will remain colleagues and I don't think anything good would come of this long-term for all involved, and perhaps even our institution. On the other hand, I'd like to make it clear to them this was wrong, that it may affect my inclination to collaborate with them in the future, and that should we collaborate, I may not be so inclined to let this slip if it happens again.