On Euler’s “Misleading Induction”, Andrews’ “Fix”, and How to Fully Automate them

Shalosh B. EKHAD and Doron ZEILBERGER^{1}

Dedicated to George Andrews on his (75 − ϵ)^{th} birthday

Recall that the trinomial coefficient([Wei])

\binom{n}{j}_{2}

is the coefficient of x^{j} in

(1 + x + x^{−1})^{n} .

In other words,

(1 + x + x^{−1})^{n} = ∑_{j=−n}^{n} \binom{n}{j}_{2}x^{j} .

Also recall that the Fibonacci numbers, F_{n}, are defined by F_{−1} = 1, F_{0} = 0, and F_{n} = F_{n−1} + F_{n−2} for n > 0.

The fascinating story of how Euler almost fooled himself into believing that

3\binom{n + 1}{0}_{2} − \binom{n + 2}{0}_{2} = F_{n}(F_{n} + 1) ,  (Leonhard)

for all n because he checked this for the nine values −1 ≤ n ≤ 7, only to find out that it fails for n = 8, leading him to record it for posterity in [E], has been told several times, including the nice ‘popular’ book by David Wells[Wel], Eric Weisstein’s extremely useful Mathworld website[Wei], and Ed Sandifer’s famous on-line MAA column[S].

In the first three sections of George Andrews’ important article [An] (that merely serve as the motivation and background for the remaining sections that talk about deep q-analogs), he describes a brilliant way to ‘correct’ the left side of of (Leonhard) in order to make the identity come true for all n ≥ −1.

First he used the obvious fact that

\binom{n + 2}{0}_{2} = \binom{n + 1}{−1}_{2} + \binom{n + 1}{0}_{2} + \binom{n + 1}{1}_{2}

(and symmetry) to rewrite Eq. (Leonhard) as:

\binom{n + 1}{0}_{2} − \binom{n + 1}{1}_{2} = \frac{1}{2}F_{n}(F_{n} + 1) ,  (Leonhard′)

1 Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen

Rd., Piscataway, NJ 08854-8019, USA. zeilberg at math dot rutgers dot edu , http://www.math.rutgers.edu/~zeilberg/ . April 3, 2013. Accompanied by Maple package GEA downloadable from Zeilberger’s website. Supported in part by the NSF.

and then went on to prove (using ad-hoc human ways) the identity

∑_{j=−∞}^{∞} \binom{n + 1}{10j}_{2} − ∑_{j=−∞}^{∞} \binom{n + 1}{10j + 1}_{2} = \frac{1}{2}F_{n}(F_{n} + 1) .  (George)

Note that for n < 8 the only non-zero summands in (George) are with j = 0.

[At the risk of giving away the punch-line, let’s remark that once conjectured, a fully rigorous proof of (George) can be obtained by checking it, à la Euler, for (to be safe) 0 ≤ n ≤ 20.]

Interlude: Even Giants make stupid conjectures

We are a little surprised that Euler could have believed, even for a second, that Eq. (Leonhard) is true for all n. Completely by hand (see [S]) Euler found a three-term linear recurrence with polynomial coefficients for \binom{n}{0}_{2} (that easily implies such a recurrence for 3\binom{n+1}{0}_{2} −\binom{n+2}{0}_{2}, both are easily found today with the Almkvist-Zeilberger algorithm [AlZ]), so he should have realized that it can’t equal F_{n}(F_{n} + 1) that satisfies a linear recurrence with constant coefficients (in other words it is what is called today a C-finite sequence, see [Z]).

Another way Euler could have easily realized that (Leonhard) is false is via asymptotics, even a very crude one. The ratio of consecutive terms on the left side of (Leonhard) obviously tends to 3 while the ratios of consecutive terms on the right side tends to ϕ^{2} = 2.61803 . . ..

The General case

With Maple (or Sage, or any computer algebra system), it is a piece of cake to generate many Euler-style cautionary tales, and Andrews-style fixes. Let’s summarize our findings by stating a general theorem, whose proof also tells you an algorithm how to compute rational generating functions for these sequences. This algorithm has been implemented in the Maple package GEA available from

http://www.math.rutgers.edu/~zeilberg/tokhniot/GEA .

Theorem: Let P (x) be any Laurent polynomial, and let

\binom{n}{j}_{P}

be the coefficient of x^{j} in P (x)^{n}.

Let k be a positive integer, and let a be an integer such that 0 ≤ a < k. Define the generalized Euler-Andrews sum to be

A(n, k, a) := ∑_{j=−∞}^{∞} \binom{n}{kj + a}_{P} .

The generating functions

f_{k,a}(t) := ∑_{n=0}^{∞} A(n, k, a)t^{n} , (0 ≤ a < k) ,

are rational functions of t, all with the same denominator, of degree k in t. They are easily computable by linear algebra.

Equivalently, the k sequences {A(n, k, a)}_{n=0}^{∞} (0 ≤ a ≤ k − 1) satisfy the same homogeneous linear recurrence equation with constant coefficients of order k (but of course with (usually) different initial conditions).

Proof: Let’s spell-out P (x)

P (x) = ∑_{i=α}^{β} c_{i}x^{i} ,

where α < β (and α may be negative, of course). Then, obviously, we have the analog of Pascal’s triangle identity:

\binom{n}{j}_{P} = ∑_{i=α}^{β} c_{i}\binom{n − 1}{j − i}_{P} .

Hence

A(n, k, a) = ∑_{i=α}^{β} c_{i}A(n − 1, k, a − i mod k) .

On the level of generating functions we get

f_{k,a}(t) := δ_{a,0} + t ∑_{i=α}^{β} c_{i}f_{k,(a−i) mod k} , (0 ≤ a < k) ,

where δ_{a,0} is 1 when a = 0 and 0 otherwise. This gives us a system of k linear equations in the k unknowns

{f_{k,0}(t), f_{k,1}(t), … f_{k,k−1}(t)} ,

that Maple can solve very fast. The fact that the (same) denominator has degree k, (and the numerators have degree k − 1) follows from Cramer’s rule.

This is implemented in procedure GA(P,x,k,t), in the Maple package GEA, that inputs a Laurent polynomial P in the variable x, a positive integer k and a variable t, and outputs a list of rational functions in t, of length k, whose (a + 1)-th entry is f_{k,a}(t). If P is symmetric (P (x) = P (1/x)) one only has to go as far as a ≤ k/2, since then f_{k,k−a}(t) = f_{k,a}(t). This (faster) case is handled by the procedure GAs(P,x,k,t).

Computerized Redux of Andrews’s man-made proof

The case P (x) = x^{−1} + 1 + x and k = 10 is the one that Andrews needed. So all you need is type

GAs(x+1+1/x,x,10,t); ,

giving all the six generating functions, whose coefficients, A(m, 10, a) (0 ≤ a ≤ 5), are given explicitly in Eq. (2.18) of [An] (reproduced in [Wei]) as expressions involving Fibonacci numbers and powers of 3. It follows from the theorem (even without actually computing the generating functions!) that Andrews’ claimed formulas can be proved rigorously by ‘just’ checking the first 20 special cases, as remarked above.

Ditto for Theorem 2.1 of [An], (also quoted in [Wei]). An empirical proof à la Euler (and now one can manage with m = 10 + 3 + 2 = 15 special cases) suffices.

But if you do not know beforehand conjectured expressions, then procedure GA can find the generating functions ab initio.

The Beauty of Programming

Even Euler and Andrews would soon get tired of doing the analogous thing for other k. Andrews also did the case k = 6 in Theorem 3.1 of [An], but we can do it for all k up to 100 (easily) and not just for P (x) = x^{−1} + 1 + x, but for any P (x). See the sample output in the front of this article

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/gea.html .

In particular, we can generate (many!) Euler-Style cautionary tales about the premature use of empirical induction, see procedure BCT and the webbook

http://www.math.rutgers.edu/~zeilberg/tokhniot/oGEA6 .

Encore: Probabilistic Interpretation

Suppose you have a (fair or) loaded die whose faces are marked with dollar amounts (some positive, some negative, some (possibly) 0), at each throw you ‘gain’ the amount on the landed face (of course if the amount is 0 you get nothing, and if the amount is negative, you have to pay). Let a(n) be the probability of breaking even after n throws. Then of course the generating function of a(n) is not rational, i.e. the sequence a(n) does not satisfy a linear recurrence with constant coefficients (on the other hand it does satisfy a linear recurrence with polynomial coefficients, easily found by the Almkvist-Zeilberger[AlZ] algorithm, implemented in the Maple package

http://www.math.rutgers.edu/~zeilberg/tokhniot/EKHAD.).

But fixing k (even a very large one, say a googol), then the related probability, let’s call it b_{k}(n) of getting an exact multiple of k (that for small n is the same as breaking even), does satisfy a linear recurrence equation with constant coefficients of order k, (equivalently, the generating function is a rational function of degree k). Ditto for the probability b_{k,a}(n) of finishing with an amount that leaves remainder a when divided by k.

References

[AlZ] Gert Almkvist and Doron Zeilberger, The Method of Differentiating Under The Integral Sign, J. Symbolic Computation 10(1990), 571-591.

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/duis.html

[An] George Andrews, Euler’s ’exemplum memorabile inductionis fallacis’ and q-Trinomial Coefficients, J. Amer. Math. Soc. 3 (1990), 653-669.

http://www.ams.org/jams/1990-03-03/S0894-0347-1990-1040390-4/S0894-0347-1990-1040390-
4.pdf

[E] Leonhard Euler Exemplum Memorabile Inductionis Fallacis, Opera Omnia, Series Prima, 15 (1911), 50-69, Teubner. Leipzig, Germany.

[S] Ed Sandifer, A Memorable Example of False Induction, MAA on-line

http://www.maa.org/news/howeulerdidit.html, 2005.

[Wei] Eric W. Weisstein, Trinomial Coefficient, From MathWorld–A Wolfram Web Resource.

http://mathworld.wolfram.com/TrinomialCoefficient.html

[Wel] David Wells, “Games and Mathematics”, Cambridge University Press, 2012.

[Z] Doron Zeilberger, The C-finite Ansatz, to appear in the Ramanujan J.

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/cfinite.html