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 Apr 21 answered Iteration of $A_{n}(q)=q^nA_{n-1} (q)$ Apr 19 comment Understanding Leibniz formula: $D^n (f g) = \sum\limits_{k=0}^{n} \binom{n}{k} D^{n-k}f D^kg$ @LukasArvidsson You are welcome. I am very glad that my answer helped you. Apr 12 revised Understanding Leibniz formula: $D^n (f g) = \sum\limits_{k=0}^{n} \binom{n}{k} D^{n-k}f D^kg$ deleted 3 characters in body Apr 12 answered Understanding Leibniz formula: $D^n (f g) = \sum\limits_{k=0}^{n} \binom{n}{k} D^{n-k}f D^kg$ Mar 26 awarded Nice Question Mar 22 awarded Nice Answer Mar 18 awarded Popular Question Feb 29 comment Finding Divisibility of Sequence of Numbers Generated Recursively @MarkusScheuer Great work. As you pointed out, there is big similarity with my question to prove the divisibility , Your method also worked in this question well to obtain $x^k$ terms because of $e^x$ terms. Really ,Your series expansion method is very powerful to analyze divisibility for similar problems . Thanks a lot for sharing . Best Regards Feb 26 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuer I have noticed the wrong typing now even if I am very careful about it. I apologize for wrong typing your last name in my previous comment. I cannot edit it because İt is disabled. If you want, I can delete it. Thanks a lot for your understanding Feb 26 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuer: I have not found any book or link that shows that relation $n^{d}\equiv P(n,d) \pmod {d}$. I just wondered if you found any link about it. Please share if you found a link. Best Regards Feb 25 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 39 characters in body Feb 25 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 32 characters in body Feb 25 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 631 characters in body Feb 25 comment Solving differential equation $y''(x)+Q(x)y(x)=0$ If you know any particular solution $y_p(x)$ that solves $y''(x)+Q(x)y(x)=0$ , You will find the general solution of your equation. There is no closed known solution method to find the particular solution if $Q(x)$ is not given. Of course there are a lot of endless series methods exist to solve the equation but no closed formula for general $Q(x)$. Feb 25 comment Solving differential equation $y''(x)+Q(x)y(x)=0$ If you do not know an particular solution $(y_1(x))$ for the equation , there is no known general solution for the equation for $Q(x)$. If you want to check my question that related to this problem. math.stackexchange.com/questions/99850/… Feb 25 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuner Your last bonus opens a huge mysterical door for new research field. I have been searching the internet for $$n^d \equiv P(n,d) \pmod {d}$$ but I have not found the theory. Have you found a related work in somewhere ? Best Regards Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 23 characters in body Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 38 characters in body Feb 23 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuer Thank you for your advice and help . I added an answer that shows the proof of my conjecture. Please advice if there is a mistake in it. Best Regards Feb 23 revised How many different numbers can be written if each used digit symbol is used at least 2 times? added 57 characters in body