In other words, in which book or article and on which page can I find the following conclusion in the picture? Thank you in advance!
On 15 January 2015, Mr De-Yu Zhai, an afterhours mathematician in China, pointed out that there are two related conclusions on pages 540--541 in the book [K. Knopp, Theory and Application of Infinite Series, Second English Edition, translated from the Fourth German Edition by Miss R. C. H. Young, Blackie & Son Limited, Glascow-Bombay-Toronto, 1951]. See pictures attahed here. However, the general formula for the general terms is not given explicitly.
On 16 January 2015, Dr Professor Cristinel Mortici in Romania e-mailed me that a related conclusion was cited in the paper [N. Elezovic And L. Vuksic, Asymptotic expansions of bivariate classical means and related inequalities, J. Math. Inequal. 8 (2014), no. 4, 707-–724; Avaialble online at http://dx.doi.org/10.7153/jmi-08-53]. See the picture below. In that paper, the following two references were cited:
[1] A. Erdelyi, Asymptotic Expansions, Dover Publications, New York, 1956.
[2] H. W. Gould, Coefficient identities for powers of Taylor and Dirichlet series, Amer. Math. Monthly 81 (1974), no. 1, 3–14. (This reference was mistakenly cited in the paper by Elezovic and Vuksic)
However, I can not find anything related to the formula in question and the one cited in the picture by Elezovic and Vuksic.
On 18 January 2015, Lenka Mihoković replied me by e-mail and recommended the following paper:
C.-P. Chen, N. Elezović, and L. Vukšić, Asymptotic formulae associated with the Wallis power function and digamma function, J. Classical Anal. 2 (2013), no. 2, 151--166; Available online at http://dx.doi.org/10.7153/jca-02-13.
In this paper, among other things, the authors proved the following conclusions. These do not answer directly and explicitly to the original question. I wonder if there have been any older references explicitly containing these conclusions and their proofs.
The following formally published paper is related to this question:
Feng Qi, Xiao-Ting Shi, and Fang-Fang Liu, Expansions of the exponential and the logarithm of power series and applications, Arabian Journal of Mathematics 6 (2017), no. 2, 95--108; available online at https://doi.org/10.1007/s40065-017-0166-4
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