An extension of Jensen's form (Q2721596)

This is a very sloppy and unreadable paper. The introduction contains already quite a number of pretty obvious errors. We start with the remark that formulas (1.1) and (1.2) are not equivalent, as suggested by the author. In stead of (1.2) he probably means the following formula: NEWLINE\[NEWLINE\sum^\infty_{r=0} {n-ar\choose r} c^r b^{n-(a+1)r}= {\zeta^{n+1}_0\over (a+1)\zeta_0- ab}.NEWLINE\]NEWLINE The last formula can easily be seen to be equivalent to (1.1) by taking \(c\), \(y\) and \(z\) as stated in the last line of page 279. Furthermore I do not trust the remark on the number of words of \(n\) letters over an alphabet of \(b\) not containing a word of \(a\) letters as made on page 280. As a first term in the partial sums of the corrected form of (1.2) \((r=0)\) contains the term \(b^n\), the other terms of the partial sum being positive gives a total that is bigger than \(b^n\), i.e. the number of words over an alphabet of \(b\) letters with restriction. So the remark obviously must be wrong. Finally putting \(a= 1\) in the corrected form of (1.2) gives: NEWLINE\[NEWLINE\sum^\infty_{r=0} {n-r\choose r} c^r b^{n-2r}= {\zeta^{n+1}_0\over 2\zeta_0- b},NEWLINE\]NEWLINE where \(\zeta_0\) is the biggest zero of \(z^2- bz- c\), i.e. \(\zeta_0= {b+\sqrt{b^2+ 4c}\over 2}\). Therefore NEWLINE\[NEWLINE\sum^{\lfloor n/2\rfloor}_{r=0} {n-r\choose r} c^r b^{n-r}= {({b+ \sqrt{b^2+ 4c}\over 2})^n\over \sqrt{b^2+ 4c}}NEWLINE\]NEWLINE would be in accordance with the corrected formula (1.2). This differs from the result in (1.3) even putting \(a=1\).NEWLINENEWLINENEWLINEGoing from (2.1) to the first formula on page 281 requires in my eyes a lot of more explanation then the author is willing to give. The same holds for the rest of the formula that seem to come right out of the blue. Finally, it is very unclear to me which of the formula's in this paper generalizes the ``correct'' form of (1.2).