6

The identity is

$$\sum_{i\ge 0}(-1)^i \binom{\frac{s^k + s^{-k} - 10}{4}}{i}\binom{\frac{s^k + s^{-k} - 10}{4}+4}{\frac{gs^k + 2g^{-1}s^{-k} - 4}{8}-i}= 0$$

where $k\gt 1$ , $s = 3+2\sqrt{2}$ and $ g = 2-\sqrt{2}$

exemples $$\sum_{i\ge 0}(-1)^i \binom{6}{i}\binom{10}{2-i}= 0$$ $$\sum_{i\ge 0}(-1)^i \binom{47}{i}\binom{51}{14-i}= 0$$ $$\sum_{i\ge 0}(-1)^i \binom{286}{i}\binom{290}{84-i}= 0$$ $$...$$ ....etc.

It comes from this question.

Are there other identities like this one, where the integers are obtained from a recursion?

A generalisation (in the neater presentation suggested by @JeanMarie) would be, for a given positive integer $p$

$$\sum_{i\ge 0}(-1)^i \binom{u_k}{i}\binom{u_k+p}{v_k-i}= 0$$

This was the case $p=4$. The cases $p\le 3$ are dealt in the previous question.

For $p=5$ we have four pairs of sequences $(u_k,v_k)$

$$u_k=18u_{k-1}-u_{k-2}+48\ \ \ v_k=18v_{k-1}-v_{k-2}+8$$ with $$u_0=2,u_1=-2\ \ \ v_0=6,v_1=0$$ $$u_0=-1,u_1=-1\ \ \ v_0=2,v_1=0$$ $$u_0=-1,u_1=-1\ \ \ v_0=3,v_1=1$$ $$u_0=-2,u_1=2\ \ \ v_0=1,v_1=3$$

For $p=6$ we have two pairs of sequences $(u_k,v_k)$

$$u_k=14u_{k-1}-u_{k-2}+42 \ \ \ v_k=14v_{k-1}-v_{k-2}+6$$ with $$u_0=-1,u_1=-2\ \ \ v_0=4,v_1=0$$ $$u_0=-2,u_1=-1\ \ \ v_0=2,v_1=0$$

For $p=8$ we have two pairs of sequences $(u_k,v_k)$

$$u_k=6u_{k-1}-u_{k-2}+18 \ \ \ v_k=6v_{k-1}-v_{k-2}+2$$ with $$u_0=-1,u_1=-2\ \ \ v_0=5,v_1=1$$ $$u_0=-2,u_1=-1\ \ \ v_0=3,v_1=1$$

Question: Find other pairs of sequences $(u_k,v_k)$, $v_k \lt u_k$ for $p=7$ or $p\ge 9$ (if any).

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René Gy
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1 Answers1

2

Two pieces of information

1) I advise you to read Chapter 5 of the marvelous book "Concrete Mathematics" (Graham, Knuth, Patashnik) Addison-Wesley, 1989, which is in the same spirit as the book "A=B" that has been recommended by @Gabriel Nivasch.

2) Your expression should be simplified in this way:

$$\sum_{i\ge 0}(-1)^i \binom{u_k}{i}\binom{u_k+4}{v_k-i}= 0$$

where $u_n$ and $v_n$ are the sequences defined resp. by the second order recurrence relationships:

$$u_{n+1}=6u_n-u_{n-1}+10 \ \ \text{with} \ \ u_0=6 \ \text{and} \ u_1=47$$

and

$$v_{n+1}=6v_n-v_{n-1}+2 \ \ \text{with} \ \ v_0=10 \ \text{and} \ v_1=51$$

In this way you have a neater presentation without spurious irrationals.

Jean Marie
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  • That is a fantastic book. What do you think is the required maturity level and prequisties before tackling Concrete Mathematics? – Saikat Mar 13 '16 at 05:40
  • @user230452 If you are ending your undergraduate studies in Maths, you can already get something out of it. But it is not a book that you read from cover to cover. The main idea it conveys is "Think hypergeometrics". Using Mathematica (or Maple) in parallel helps a lot. – Jean Marie Mar 13 '16 at 05:48
  • I am an undergraduate student in CS altough I really like Mathematics more, which I spend nearly all my time doing. Can I get a lot out of this book or how long do I need to wait? I've been wanting to read that book for a while now. – Saikat Mar 13 '16 at 06:15
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    @user230452 You can read it from www.csie.ntu.edu.tw/~r97002/temp/Concrete%20Mathematics%202e.pdf The question about when is it preferable to read a book has no true answer. It's true that sometimes, preliminary lectures or textbooks may be preferable. There are some books like this one that are especially interesting for people like you, at the boundary between CS and Maths. Of course, there are the books of Knuth ("The Art of Computer Programming"), "Gödel, Escher, Bach" (D. Hofstadter), "Computational Geometry" (De Berg, Van Kreveld, Overmars, Schwartzkopf), etc... – Jean Marie Mar 13 '16 at 06:39
  • I hope to switch over to Mathematics someday. Thanks for the link, but I already have it. Unfortunately, I find AOCP too hard at the moment and I'm targetting to read and Finish Concrete Mathematics in my fifth(next) semester. But, your comment about requiring an undergraduate degree in Maths made me kind of nervous. I'll check out your third recommendation shortly! Thanks! And wish me luck in working through that book by the end of this year! – Saikat Mar 13 '16 at 07:04
  • @JeanMarie yes I know of the neater presentation, I wanted to be a bit provocative :-) to try to tease some interest. Thanks for the reference to the book, I didn't know it can be freely read. – René Gy Mar 13 '16 at 10:29