@Axel Vogt  i just put this conditions for  a positive integer number, but i need to definie a funtion (sequence of function depend of 't' not m , m just indexation)

@tomleslie i'm so thankful for you, i am doing a work i hope that is work but o found the problem in affectation of functions psii (sequence of function psii_m(t)) with x it depend with m and t and i want it to based at last step for t to define my integral i was in a variable transformation of trapeze rule with x=psi(t) and psi(t)= theta(t)/thata(1) and any step i change m. ok i'll test your solution, thank you so much, test_tr.mw
Extension_of_a_class_of_periodizing_variable.pdf   (i work with this article and i work with numerical example) is that why i have encountred some errors.

@tomleslie this is my codes, one see test.mwv thanks my friend
test_tr.mw

or here 

[1]: https://www.dropbox.com/s/6f3j577u2ikh7gk/test%20tr.mw?dl=0
if any one can help me.

@tomleslie i don't know how to insert the code but i need a help if you can help me i have a show tomorrow and i must give some numerical examples and this one must work with me , my e-mail is : abdelkarimsm@gmail.com. thank's a lot for your help i'm so thankful

@tomleslie  i want to use 'm' as andex, and t as a simple variable, i need to use that because with this recursive sequence i want to compute an integral basing from this recursive sequence of function if you can help me in that i'm so thankful for your comment @tomleslie