Dear Feng,

See some computations for A2n.

Look the factor analysis in prime numbers of those coefficients.

They seem to have as a common factor the number 3.

I hope to find a kind of help

Demetris

Dear Demetris Christopoulos

What you found is very very interesting for me. I need a general formula for the coefficients $A_{2n}$. Could you please further help me?

Best regards

Feng Qi (F. Qi)

The common factor $3$ comes from the power $3$. All the derivatives of the function $[f(t)]^k$ must contain a factor $k$, mustn't it?

It is a difficult work to find a closed form for A2n. That's why I tried to see if there exists an obvious sequential form. Since I am in vacations now, I cannot do the detailed computations needed. Later I will write a general guide for such problems, at least in order to be able to find the first k coefficients. You can look up series theory in complex analysis, because the problem belongs there. As for the second question, it is true for $k\geq{1}$.

Suppose sec3  z = a0+ a2 z2 + a4z 4 + .........                                                                                  Now cos3 z = ( cos 3z + 3 cos z) / 4 =b0 + b2 z2 + b4z4 + .......      where b0, b2 , b4 ,...etc are known because series for cos z is known. Therefore,

        a0+ a2 z2 + a4 z 4 + .........  = 1. / (b0 + b2 z2 + b4 z4 + .......  )

or (  a0+ a2 z2 + a4 z 4 + .........  )(b0 + b2 z2 + b4 z4 + .......  ) =1.

or a0b0  = 1.,  a0b2 + a2b0 = 0, a0b4+a2b2 + a4b0 =0    and so on

   From here a0, a2, a4 etc can be obtained.

Dear Mittal

Can you obtain a closed form for ak in terms of Euler numbers or other famous numbers? As the secant does. In other words, can you obtain an explicit and general formula for all ak? It seems not easy for me.

Sincerely yours

Feng QI

Dear Dr. Feng,

     Kindly see the attachment for explicit formula where a2r can be obtained from previous known values  a2r - 2 etc. I am not able to connect to Euler's number at the moment.

Thank Mittal very much. But I can not read your attachment regularly, because it looks like the picture below

Please find attachment in pdf format.

Dear Dr. Feng,

Take a look at:

Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1

in particular section: 7.1 Multinomial Euler numbers

The results in section 7.1 are for powers of sech, so you'll need to substitute t -> it.

The above paper is freely available from the Journal's website, and also from my ResearchGate pages.

Best wishes,

Ghislain Franssens

[email protected]

There was typo error in my pdf file. Attached is the correct one.

Thank Ghislain R. Franssens and R. C. Mittal. I'll come back here a few days later.

Feng

It is quite easy to see that if

\begin{equation}

\sec^3z=\sum_{n=0}^\infty A_{2n}\frac{z^{2n}}{(2n)!}, \quad |z|

Please, Feng Qi could you clarify your question:

Do you mean the series expansion of Sec3(x) at x = 0 ? 

Dear Amaechi J. Anyaegbunam,

you are right. My question is just to obtain the power series expansion of the funciton sec3x at x=0. I am looking forward to explicit, easily computable, closed expression of coefficients A2n. Thank for your interest.

Best regards

Feng Qi (F. Qi)

Dear Feng

This is my answer obtained via Maclaurin's series

 Sec3x  = 1 + (3/2)*x2 +(11/8)*x4 + (241/240)*x6 + (8651/13440)*x8+.....                                                                                                                                                               which is accurate for x < = 0.5. Unfortunately, this series does not exhibit                    any regularity to enable its extension to higher powers of x.                                         I will attempt to add two more terms or alternatively device another procedure              for the expansion.                           

Dear Amaechi J. Anyaegbunam, the result you showed me may be easily obtained by the software Mathematica or by any of other famous softwares. What I need is a closed form for the coefficient of the general term x2k in the power series expansion of the function sec3x. What is a closed form? Please click the link http://en.wikipedia.org/wiki/Closed-form_expression.

Since there is a no closed form expression for the Euler numbers, it is not expected to obtain a closed-form expression for the coefficients of the Sec^3 expansion.

One may expect some recursive algorithms as for instance that for the Euler numbers, e.g.

MAPLE

P := proc(n, x) option remember; if n = 0 then 1 else

(n*x+(1/2)*(1-x))*P(n-1, x)+x*(1-x)*diff(P(n-1, x), x);

expand(%) fi end:

A122045 := n -> (-1)^n*subs(x=-1, P(n, x)):

seq(A122045(n), n=0..30); # Peter Luschny, Mar 07 2014

cp. the first link for the Euler (or secant) numbers given below.

See also the second link below for another algorithm for the secant numbers.

http://oeis.org/A122045

http://arxiv.org/pdf/1108.0286v3.pdf

Dear Feng

Could you modify your question to clarify what you want. Your current question is             "What is the power series expansion at zero of the secant to the power of three?"

It should read "What is the closed-form expression for the coefficients of the power series expansion of secant(x) to the power of three at zero? "  

Thank you!    

Is there not a closed form expression for Euler numbers? Is the formula showed in the following picture a closed form expression for Euler numbers? This formula was obtained in the following article:

Bai-Ni Guo and Feng Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, Journal of Computational and Applied Mathematics 272 (2014), 251--257; Available online at http://dx.doi.org/10.1016/j.cam.2014.05.018. Click also the site at http://qifeng618.wordpress.com/2014/07/05/a-link-to-freely-download-one-of-my-articles/

http://qifeng618.wordpress.com/2014/07/05/a-link-to-freely-download-one-of-my-articles/

http://dx.doi.org/10.1016/j.cam.2014.05.018

well if you have a closed form expression, why not sent it to the link that I gave before?

If you now have closed form expression for the coefficients of the third power of a series in terms of the coefficients of the original series, you just have to put in the secant number expression, and your question is answered :)

I now see that the Euler number you mean is not the Euler number I mean. You mean the secant number, I mean the Euler number. Sometimes they are all called the Euler number. Euler numbers have closed form expressions, but the secant numbers have not. As Herbert H. H. Homeier said, since there is no closed form expression for the secant numbers, it is not expected to obtain a closed-form expression for the coefficients of the Sec^3 expansion. Thank Herbert H. H. Homeier!

As far as I know (see also the links given previously), Euler numbers and secant numbers are used interchangebly. Actually, what do you mean exactly by the Euler numbers? Does that use  fit to the link A122045 - OEIS as given above?

Dear Herbert H H Homeier, you are right:  Euler numbers and secant numbers are used interchangebly. They may be defined by two manners: Picture 1 and Picture 2. In other words, Euler numbers and secant numbers are the same sequence.

Now I have a question to ask for your confirmation again: Is there no closed form expression for the secant (Euler) numbers? Who said that there is no closed form expression for the secant (Euler) numbers? Where to look up the conclusion that there is no closed form expression for the secant (Euler) numbers? As I said above, I obtained a formula for Euler numbers in terms of Stirling numbers, see Picture 4. Is this formula a closed form expression for the secant (Euler) numbers in your opinion? Feng Qi (F. Qi)

I just repeat: If you have a closed form expression ...

The question then is: do you have a closed form for the Stirling numbers?

:)

Stirling numbers S(n,k) may be computed by a finite sum: see Picture 5. Is it a closed form expression for Stirling numbers? I am not sure.

Just now I searched out a reference at http://arxiv.org/abs/1103.1585. From the viewpoint of this preprint, closed form expressions for secant (Euler) numbers surely exist.

It is much difficult for me to read the link http://oeis.org/A122045.

See http://oeis.org/eishelp2.html

Assume that secant numbers are finitely expressed in closed form. Then coefficients of m-th powers of the secant powers series are finitely expressed by using (m-1)-fold convolutions of the sequences of secant numbers (cf. http://en.wikipedia.org/wiki/Power_series#Multiplication_and_division), e.g. the k-th coefficient of the third power is given by

(E*E*E)_k = \sum_{m=0}^k E_{k-m} \sum_{n=0]^m  E_{m-n} E_n

Beware that here, the odd secant numbers are simply 0.

http://en.wikipedia.org/wiki/Power_series#Multiplication_and_division

Dear All, How are you going? I have a good news to tell you: Several days ago I found a method to give a closed form for coefficients of MacLaurin series of the function $(sec z)a$, where $a$ may be any given number. When I complete themanuscript, I would announce here.

This problem is essentially a problem that how we can get a general formula for the derivatives of the function (sec z)a. It may be realized by virtue of the Fa\'a di Bruno formula. These days I lost interest to this problem, so till now I did not show the answer for this problem. Pardon me.

For detailed information, please read the preprint at http://arxiv.org/abs/1202.1205.

The following formally published papers are related to this question:

[1] Feng Qi, Derivatives of tangent function and tangent numbers, Applied Mathematics and Computation 268 (2015), 844--858; available online at http://dx.doi.org/10.1016/j.amc.2015.06.123

[2] Feng Qi and Miao-Miao Zheng, Absolute monotonicity of functions related to estimates of first eigenvalue of Laplace operator on Riemannian manifolds, International Journal of Analysis and Applications 6 (2014), no. 2, 123--131.

[3] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1