\magnification=\magstep0 \documentstyle{amsppt} \pagewidth{6.2in} \input amstex \topmatter \title $q$-Ap\'ery Irrationality Proofs by $q$-WZ pairs \endtitle \author Tewodros Amdeberhan and Doron Zeilberger \endauthor \affil Department of Mathematics, Temple University, Philadelphia PA 19122, USA \\ tewodros\@math.temple.edu, zeilberg\@math.temple.edu \endaffil \abstract Using WZ pairs, Ap\'ery-style proofs of the irrationality of the q-analogues of the Harmonic series and $Ln(2)$ are given. For the q-analogue of $Ln(2)$, this method produces an improved irrationality measure. \endabstract \endtopmatter \def\({\left(} \def\){\right)} \def\R{\bold R} \document \bf 0. Introduction: \rm \smallskip Let us define the following q-analogues of the Harmonic series $\sum_{n=1}^{\infty}\frac 1n$ and Ln(2), respectively by: $$\alignat2 h_q(1):&=\sum_{k=1}^{\infty}\frac 1{q^k-1} &&\qquad\text{(for $|q| > 1$),} \tag0.1 \endalignat$$ $$\alignat2 Ln_q(2):&=\sum_{n=1}^{\infty}\frac {(-1)^n}{q^n-1} &&\qquad\text{(for $|q| \neq 0, 1$)}.\tag0.2 \endalignat$$ In 1948, Paul Erd\"os [E1] proved the irrationality of $h_2(1)$. Recently, Peter Borwein used Pad\'e approximation techniques [B1] and some complex analysis methods [B2] to prove the irrationality of both $h_q(1)$ and $Ln_q(2)$. Here we present a proof in the spirit of Ap\'ery's magnificent proof of the irrationality of $\zeta(3)$ [A], which was later delightfully accounted by Alf van der Poorten [P]. This method of proof gives favorable irrationality measure (=4.80) for $Ln_q(2)$ campared to the irrationality measure (=54.0) implied in [B1], [B2]. Further discussion of irrationality results for certain series is to be found in Erd\"os [E2]. \smallskip We will assume familiarity with ref. [Z]. In particular, \par $\binom{n}k_q := \frac {(q)_n}{(q)_k (q)_{n-k}}$, where $(q)_0:=1$ and $(q)_n: = (1-q) \cdots (1-q^n),$ for $n \geq 1.$ \smallskip N and K are \it forward shift operators \rm on $n$ and $k$, respectively. \smallskip $\Delta_n := N - 1$, $\Delta_k := K - 1$. \smallskip A pair $(F(n,k),G(n,k))$ of discrete functions is called a \it q-WZ pair \rm if:\par 1. $NF/F$, $KF/F$, $NG/G$ and $KG/G$ are all rational functions of $q^n$ and $q^k$, and \par 2. $\Delta_n F = \Delta_k G.$\smallskip Given such a pair $(F,G)$, then $\omega = F(n,k) \delta k + G(n,k)\delta n$ is called a \it q-WZ 1-form. \rm \pagebreak \midspace{.3in} \bf 1. A scheme for proving the irrationality of the \it q-harmonic series \bf $h_q(1)$:\rm \bigskip The claims made in subsections \bf 1.1-1.5 \rm below were found using the Maple Package {\tt qEKHAD} accompanying [PWZ]. The relevant script substantiating our claims can be found in this paper's Web Pages.\smallskip \bf 1.1. \rm The q-WZ 1-form $\omega$ is: $$\omega = \frac {-1}{\binom {n+k+1}{k}_{q}(q)_{n+1}} \{\delta k + \frac {q^{n+1}}{(q^{n+1}-1)}\delta n\}.$$ \bf 1.2. \rm The choice of the potential $c(n,k)$ is: $$c(n,k)=\sum_{m=1}^{n}\frac {q^m}{(1-q^m)(q)_{m}} + \sum_{m=1}^{k}\frac 1{(q^m-1)}\frac 1{\binom {n+m}m_{q}(q)_{n}}.$$ \smallskip \bf 1.3. \rm The choice of the mollifier $b(n,k)$ is: $$b(n,k)=(-1)^k q^{k(k+1)/2}\binom {n+k}k_{q}\binom nk_{q}.$$ \bf 1.4. \rm We define two sequences: $$ a(n)=\sum_{k=0}^{n}c(n,k)b(n,k), \qquad\text{and} \qquad b(n)=\sum_{k=0}^{n}b(n,k).$$ \bf 1.5. \rm Introduce $L=y_2(n)N^2+y_1(n)N+y_0(n)$ and $B(n,k) = P_q^1(n,k) b(n+1,k)$, where $$ A(n,k)=c(n,k)B(n,k)+\frac{(-1)^kq^{2n+3}}{q^{n+1}-1}\binom {n+1}k_q\frac {q^{\binom k2}}{(q)_{n+2}}P_q^2(n,k) \qquad\text{and}$$ $P_q^1(n,k)=-q\alpha_n^2\beta_k^{-1}(q^2\alpha_n+2q)+q\alpha_n^2(q^2\alpha_n^3+ 2q(q+1)\alpha_n^2+3q\alpha_n-(q+1)-(\alpha_n+2)\beta_k)$ $$\multline P_q^2(n,k)=q^2\alpha_n^2+q\alpha_n-2+\beta_k(q^2\alpha_n^5+q(2q+1)\alpha_n^4-2\ alpha_n^3-\alpha_n^3\beta_k-(2-q^{-1})\alpha_n^2\beta_k\\ -(3q+5)\alpha_n^2+2q^{-1}\alpha_n\beta_k+(q-1+2q^{-1})\alpha_n+(1+3q^{-1})), \endmultline$$ $y_0(n)= q(\alpha_n-1)(q\alpha_n+2)$, ${ }$ $y_2(n)=(q\alpha_n-1)(\alpha_n+2)$, $ {}$ $\alpha_n=q^{n+1}$, $ { }$ $\beta_k=q^{k+1}$ and \bigskip $y_1(n)=q^3\alpha_n^5+2q^2(q+1)\alpha_n^4 +q^2\alpha_n^3-4q(q+1)\alpha_n^2 +(q^2-4q+1)\alpha_n+2(q+1).$ \bigskip Then $$\alignat3 L(b(n,k))&=B(n,k)-B(n,k-1) &\qquad\text{and} &\qquad L(b(n,k)c(n,k))&=A(n,k)-A(n,k-1).\tag{$*$}\endalignat$$ Now, summing over $k$ in $(*)$ shows that both sequences $a(n)$ and $b(n)$ are solutions of $Lu(n) = 0$. \pagebreak \midspace{.4in} \bf 1.6. \rm Set $b_n=b(n)$ and $a_n=a(n)$. Now, since $b_{n+1} > b_n$ and $Lb_n = 0$, that is, \par $y_2(n)b_{n+2}+y_1(n)b_{n+1}+y_0(n)b_n = 0$, then asymptotically we have that $$\frac {b_{n+2}}{b_{n+1}} = O\(\frac{y_1(n)}{y_2(n)}\) = O\(q^{3n+3}\).$$ Hence, $$b_n = O\(q^{\frac{3n^2}2}\).\tag1.6.1$$ On the other hand, $La_n = 0$ and $Lb_n = 0$ lead to the system of recurrence relations, $$\alignat2 y_2(n)a_{n+2}+y_1(n)a_{n+1}+y_0(n)a_n = 0,&\qquad y_2(n)b_{n+2}+y_1(n)b_{n+1}+y_0(n)b_n = 0.\tag1.6.2\endalignat$$ Multiplying out the first and the second equations in (1.6.2), respectively by $b_{n+2}$ and $a_{n+2}$, and subtracting we obtain $$y_1(n)(a_{n+1}b_{n+2}-b_{n+1}a_{n+2}) = y_0(n)(a_{n+2}b_{n}-b_{n+2}a_{n}).$$ Rewriting this in the form $$\frac {a_{n+1}}{b_{n+1}}- \frac {a_{n+2}}{b_{n+2}} = \frac{y_0(n)}{y_1(n)}\frac {b_n}{b_{n+1}}\(\frac{a_{n+2}}{b_{n+2}}- \frac {a_{n}}{b_{n}}\)$$ leads to the estimate $$\left|\frac {a_{n+1}}{b_{n+1}}- \frac {a_{n+2}}{b_{n+2}}\right| \leq \left|\frac{y_0(n)}{y_1(n)}\frac {b_n}{b_{n+1}}\(\frac{a_{n+2}}{b_{n+2}}- \frac {a_{n+1}}{b_{n+1}}\)\right|+ \left|\frac{y_0(n)}{y_1(n)}\frac {b_n}{b_{n+1}}\(\frac{a_{n+1}}{b_{n+1}}- \frac {a_{n}}{b_{n}}\)\right|,$$ which in turn yields $$\frac {a_{n+1}}{b_{n+1}}- \frac {a_{n}}{b_{n}} = O\(b_n^{-2}\).\tag1.6.3$$ Therefore, $$h_q(1)-\frac {a_n}{b_n}=O\(b_n^{-2}\).\tag1.6.4$$ In particular, the sequence of rational numbers $\frac {a_n}{b_n}$ converges moderately quickly to $h_q(1)$. \pagebreak \midspace{.3in} \bf 1.7. \rm For a given prime $p$, let $ord_p$k denote the exponent of $p$ in the prime expansion of $k$. Then we observe that $$ord_p \binom nm_q \leq ord_p(q)_n - ord_p(q)_m.\tag1.7.1$$ \bf Note: \rm $$\binom{n+k}k_q\binom km_q = \binom{n+m}m_q\binom{n+k}{k-m}_q.\tag1.7.2$$ \bf Lemma 1: \rm The sequences $$ u_n=a_n(q)_{n+1}\prod_{s=[n/2]}^n(1-q^s) \qquad\text{and} \qquad z_n= b_n(q)_{n+1}\prod_{s=[n/2]}^n(1-q^s)$$ are polynomials in $q$ with integer coefficients, and moreover $$z_n = O\(q^{19n^2/8}\).\tag1.7.3$$ \bf Proof: \rm Applying (1.7.1) and (1.7.2), we can estimate the denominator of $u_n$ as: $$\align ord_p\(\frac {(q^m-1)(q)_{n}\binom{n+m}m_q}{\binom{n+k}k_q}\) &\leq ord_p\(\frac {(q^m-1)(q)_{n}\binom km_q}{\binom{n+k}{k-m}_q}\)\\ &\leq ord_p(q)_{n} + ord_p(q^m-1) + ord_p(q)_k - ord_p(q)_m\\ &\leq ord_p(q)_{n} + ord_p\prod_{s=[n/2]}^n(1-q^s) + ord_p(q)_k - ord_p(q)_m\\ &\leq ord_p\((q)_{n}\prod_{s=[n/2]}^n(1-q^s)\),\endalign$$ since $m \leq k \leq n$. This proves the claim on $u_n$. And (1.7.3) follows from (1.6.1). The rest is trivial. \bigskip \bf Lemma 2: \rm $h_q(1)$ - $\frac {u_n}{z_n}$ = $O\(\frac 1{z_n^{1+\delta}}\)$; where $\delta =0.26316\dots$$>$ 0.\bigskip \bf proof: \rm From (1.6.1), (1.6.4) and (1.7.3), we gather that $$ h_q(1) - \frac{u_n}{z_n}=O\(b_n^{-2}\) =O\(q^{-3n^2}\) =O\(z_n^{-1-(5/19)}\).$$ \smallskip Thus, we have proved:\smallskip \bf Theorem 1: \rm If $|q|>1$ is an integer, $h_q(1)$ is irrational with irrationality measure 4.80. \smallskip \bf Remark 1: \rm By invoking Theorem 7 ([Z], p.596) with $\omega$ as in 1.1, we obtain the series acceleration: $$ h_q(1)=\sum_{n=1}^{\infty}\frac {q^n}{(1-q^n)(q)_n} \qquad\text{and}\qquad h_q(1)=\sum_{n=1}^{\infty}\frac {1-q^n-q^{2n}}{(q^n-1)\binom {2n}n_q (q)_n}.$$ \pagebreak \midspace{.3in} \bf 2. A scheme for proving the irrationality of $Ln_q(2)$: \rm \bigskip The claims made in subsections \bf 2.1-2.5 \rm below were found using the Maple Package {\tt qEKHAD} accompanying [PWZ]. The relevant script substantiating our claims can be found in this paper's Web Pages.\smallskip \bf 2.1. \rm The qWZ 1-form $\omega$ is: $$\omega = \frac {(-1)^k}{(1-q^{k+1})}\frac {(q)_n}{\binom {n+k+1}{k+1}_{q}(q^2)_{n}} \{\delta k + \frac {q^{n+1}}{(1+q^{n+1})}\delta n\}.$$ \bf 2.2. \rm The choice of the potential $c(n,k)$ is: $$c(n,k)=\sum_{m=1}^{n}\frac {q^m(q)_m}{(1-q^m)(q^2)_{m}} + \sum_{m=1}^{k}\frac {(-1)^{m-1}}{(1-q^m)}\frac {(q)_n}{\binom {n+m}m_{q}(q^2)_{n}}.$$ \smallskip \bf 2.3. \rm The choice of the mollifier $b(n,k)$ is: $$b(n,k)=q^{k(k+1)/2}\binom {n+k}k_{q}\binom nk_{q}.$$ \bf 2.4. \rm We define two sequences: $$ a(n)=\sum_{k=0}^{n}c(n,k)b(n,k), \qquad\text{and} \qquad b(n)=\sum_{k=0}^{n}b(n,k).$$ \bf 2.5. \rm Introduce $L=y_2(n)N^2+y_1(n)N+y_0(n)$ and $B(n,k) = P_q^1(n,k) b(n+1,k)$, where $$ A(n,k)=c(n,k)B(n,k)+\frac{(-1)^kq^{2n+3}}{1-q^{n+1}}\binom {n+1}k_q\frac {q^{\binom k2}(q)_{n+1}}{(q^2)_{n+1}}P_q^2(n,k)\qquad\text{and}$$ $$\multline P_q^1(n,k)=q\alpha_n^2\bigl[q^3\alpha_n^5+q^2(1+q)\alpha_n^4+2q(1+q^2)\alpha_n^ 3-(1-q+q^2)\alpha_n-3(1+q)\big]\\ +q\alpha_n^2\bigl[q\beta_k^{-1}(q^2\alpha_n^3+q(1+q)\alpha_n^2+(2-q)\alpha_n-2) +(q\alpha_n^3+(q-1)\alpha_n^2+(2q-1)\alpha_n)\alpha_k-2\bigr] \endmultline$$ $$\multline P_q^2(n,k)=q^2\alpha_n^3+q(1+q)\alpha_n^2+(2+q)\alpha_n+2-\alpha_n\alpha_k^2 \bigl[\alpha_n^3+(1-q^{-1})\alpha_n^2+(2-q)\alpha_n-2q^{-1}\bigr]\\ -\alpha_k\bigl[q^2\alpha_n^6+q(1+q)\alpha_n^5+(2+q+2q^2)\alpha_n^4+(1+q) \alpha_n^3+2\alpha_n^2-(2+q+q^{-1})\alpha_n+(q^{-1}-1)\bigr],\endmultline$$ $y_0(n)= -q(\alpha_n-1)(\alpha_n+1)(q^2\alpha_n^2+q\alpha_n+2)$, ${ }$ $y_2(n)=-(q\alpha_n-1)(q\alpha_n+1)(\alpha_n^2+\alpha_n+2)$, $ {}$ $$ y_1(n)=q^4\alpha_n^7+q^2(1+q)(q\alpha_n^6+\alpha_n^4)+q(1+q+q^2)(2q\alpha_n^5+\ alpha_n^3)-(1+3q+3q^2+q^3)\alpha_n^2-(1+q^2)(2+\alpha_n),$$ and $\alpha_n=q^{n+1}$, $ { }$ $\beta_k=q^{k+1}.$ \bigskip Then $$\alignat3 L(b(n,k))&=B(n,k)-B(n,k-1) &\qquad\text{and} &\qquad L(b(n,k)c(n,k))&=A(n,k)-A(n,k-1).\tag{$**$}\endalignat$$ Now, summing over k in $(**)$ shows that both sequences $a(n)$ and $b(n)$ are solutions of $Lu(n) = 0$. \pagebreak \midspace{.4in} \bf 2.6. \rm Similar arguments and estimates as in (1.6) above lead to $$Ln_q(2)-\frac {a_n}{b_n}=O\(b_n^{-2}\).\tag2.6.1$$ In particular, the sequence of rational numbers $\frac {a_n}{b_n}$ converges moderately quickly to $Ln_q(2)$. \bf 2.7. \rm \bf Lemma 3: \rm The sequences $$ v_n=a_n\prod_{t=1}^n(1+q^t)\prod_{s=[n/2]}^n(1-q^s) \qquad\text{and} \qquad w_n= b_n\prod_{t=1}^n(1+q^t)\prod_{s=[n/2]}^n(1-q^s)$$ are polynomials in $q$ with integer coefficients, and moreover $$w_n = O\(q^{19n^2/8}\).\tag2.7.1$$ \bf Proof: \rm Applying (1.7.1) and (1.7.2), we have estimates for the denominator of $v_n$: $$\align ord_p\(\frac {(1-q^m)(q^2)_{n}\binom{n+m}m_q}{\binom{n+k}k_q(q)_n}\) &\leq ord_p\(\frac {(q^m-1)(q^2)_{n}\binom km_q}{\binom{n+k}{k-m}_q(q)_n}\)\\ &\leq ord_p\(\frac{(q^2)_n}{(q)_{n}}\) + ord_p(q^m-1) + ord_p(q)_k - ord_p(q)_m\\ &\leq ord_p\(\frac{(q^2)_n}{(q)_{n}}\) + ord_p\prod_{s=[n/2]}^n(1-q^s) + ord_p(q)_k - ord_p(q)_m\\ &\leq ord_p\(\prod_{t=1}^n(1+q^t)\prod_{s=[n/2]}^n(1-q^s)\),\endalign$$ since $m \leq k \leq n$. This proves the claim on $v_n$. And (2.7.1) follows from (1.6.1). The rest is trivial. \bigskip \bf Lemma 4: \rm $Ln_q(2)$ - $\frac {v_n}{w_n}$ = $O\(\frac 1{w_n^{1+\delta}}\)$; where $\delta =0.26316\dots$$>$ 0.\bigskip \bf proof: \rm Combining (1.6.1), (2.6.1) and (2.7.1), we find that $$ Ln_q(2) - \frac{v_n}{w_n}=O\(b_n^{-2}\) =O\(q^{-3n^2}\) =O\(w_n^{-1-(5/19)}\).$$ \smallskip Thus, we have proved:\smallskip \bf Theorem 2: \rm If $|q|\neq 0, 1$ is an integer, $Ln_q(2)$ is irrational with irrationality measure 4.80.\smallskip \bf Remark 2: \rm We invoke Theorem 7 ([Z], p. 596) with $\omega$ as in 2.1, to get the accelerated series: $$ Ln_q(2)=\sum_{n=1}^{\infty}\frac {q^n(q)_n}{(1-q^n)(q^2)_n} \qquad\text{and}\qquad Ln_q(2)=\sum_{n=1}^{\infty}\frac {(-1)^{n-1}(q)_n(1-q^{3n})}{(1-q^n)^2\binom {2n}n_q (q^2)_n}.$$ \pagebreak \midspace{.3in} \Refs \widestnumber\key{PWZ} \ref\key A \by R. Ap\'ery \paper \it Irrationalit\`e de $\zeta(2)$ et \rm$\zeta(3)$ \jour Asterisque \vol 61\yr1979 \pages11-13 \endref \smallskip \ref\key B1 \by P. Borwein \paper \it On the irrationality of $\sum 1/(q^n+r)$ \rm \jour J. Number Theory \vol 37 \yr1991 \pages253-259 \endref \smallskip \ref\key B2 \by P. Borwein \paper \it On the irrationality of certain series \rm \jour Proc. Camb. Phil. Soc. \vol 112 \yr1992 \pages141-146 \endref \smallskip \ref\key E1 \by P. Erd\"os \paper \it On arithmetical properties of Lambert Series \rm \jour J. Indian Math. Soc. (N.S.) \vol 12 \yr1948 \pages63-66 \endref \smallskip \ref\key E2 \by P. Erd\"os \book \it On the irrationality of certain series: problems and results \rm \publ In New Advances in Transcendence Theory (Cambridge University Press) \yr1988 \pages102-109 \endref \smallskip \ref\key P \by A. van der Poorten \paper \it A proof that Euler missed ..., Ap\'ery's proof of the irrationality of \rm $\zeta(3)$ \jour Math. Intel. \vol 1 \yr1979 \pages195-203 \endref \smallskip \ref\key PWZ \by M. Petkov\v sek, H.S. Wilf, D. Zeilberger\book \it ``A=B'' \rm \publ A.K. Peters Ltd. \yr1996 \endref {\tt The package qEKHAD is available by the www at http://www.math.temple.edu/\~{}zeilberg/programs.html} \smallskip \ref\key Z \by D. Zeilberger \paper \it Closed Form (pun intended!) \rm \jour Contemporary Mathematics \vol 143 \yr1993 \pages579-607 \endref \endRefs \enddocument