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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.
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
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
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
comment How many different numbers can be written if each used digit symbol is used at least 2 times?
@MarkusScheuer : Thank you a lot for your answer. It gave me a great point how to find the general formula of $P(n,d)$. Now I am testing it for some values. Best Regards
Feb
23
comment How many different numbers can be written if each used digit symbol is used at least 2 times?
@MarkusScheuner : And then easily $A_n(x)=(e^x - x)^n$ can be gotten. It is very nice result
Feb
23
comment How many different numbers can be written if each used digit symbol is used at least 2 times?
@MarkusScheuner : Wonderful relations . Please check the relation while applying the recurrence relation in (4) . I think it should be $$\sum_{d=0}^\infty (P(n+1,d)+d.P(n,d-1))\frac{x^d}{d!}\\$$. then we can get $$A_{n+1}(x)=(e^x-x)A_{n}(x)$$
Feb
21
comment How many different numbers can be written if each used digit symbol is used at least 2 times?
@MarkusScheuner : I believe that $P(n,d)$ function can be a key function to detect prime numbers. It has wonderful features
Feb
21
comment How many different numbers can be written if each used digit symbol is used at least 2 times?
Your answer is great. Have you read my conjecture : $n^{d}\equiv P(n,d) \pmod {d}$ , I have not found any counter-example yet. It satisfies for any number in your table but I do not know how to prove it .Thanks for helps.
Feb
21
comment How many different numbers can be written if each used digit symbol is used at least 2 times?
:Thanks a lot for answer . I edited my question with wonderful discover for $P(n,d)$. Could you please share your comments on my last edit?
Feb
21
comment How many different numbers can be written if each used digit symbol is used at least 2 times?
Thanks a lot for answer . I edited my question with wonderful discover for $P(n,d)$. Could you please share your comments on my last edit?
Jan
15
comment What is lower limit condition of a surface of a tetrahedron?
Can we have only one lower condition ? or impossible. If it exists, it should be cyclic .for example, the lower condition could be $|S_2-S_3|+|S_3-S_4|+|S_2-S_4|<S_1$ but I do not know if it exists such cyclic condition or how to prove that it is impossible to eliminate 3 necessary condition into one condition.
Jan
5
comment $e^x(\ln x-c) =\sum \limits_{k=0}^\infty \frac{ x^{k} \Gamma'(k+1)}{ (k!)^2}$ Is it correct result?
I have asked a question in math.overflow that is related to your computation a long time ago. mathoverflow.net/questions/227642/… . I saw your comment about your result. Thanks a lot for your comment. It is very supporting comment but the question is voted as off-topic. . I am sure that it is very interesting and helpful topic question. Could you please edit my question in overflow for more formal mathematical perspective? Thanks
Nov
4
comment Show $F(z)=\int_{0}^{1}{g(t)\over t-z}dt$ is holomorphic in $\Bbb{C}\setminus[0,1]$. Limit problem.
Please check Leibniz integral rule. The proof has similiar idea. en.wikipedia.org/wiki/Leibniz_integral_rule#Proofs Then have a look the title 'General form with variable limits' in the page
Nov
4
comment Show $F(z)=\int_{0}^{1}{g(t)\over t-z}dt$ is holomorphic in $\Bbb{C}\setminus[0,1]$. Limit problem.
The integral depends on $t$ but we take limit on $h$.
Sep
21
comment (Beginner) Intuition to solve a functional equation and steps for this particular-
@Chappers Thanks for those points