# Generalizing identities for Pi & its convergents

Generalizing identities for Pi & its convergents

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Lucas in his publication found, by brute-force search – using Maple programming, several different variants of integral identities which relate each of several first Pi convergents&lt;para&gt;(described in terms of OEIS sequences as A002485(n)/A002486(n)) to Pi.&lt;/para&gt; &lt;para&gt;I conjecture the following identity below, which represents a generalization of Stephen Lucas’ experimentally obtained identities:&lt;/para&gt; &lt;para&gt;(-1)^n*(π-A002485(n)/A002486(n))= 1/abs(l)*2^j)*Integrate(x^m*(1-x)^m*(k+(k+l)*x^2)/(1+x^2),x= 0..1)&lt;/para&gt; &lt;para&gt;where integer n = 0,1,2,3,… serves as index for terms in OEIS A002485(n) and A002486(n),&lt;/para&gt; &lt;para&gt;and {i, j, k, l, m} are some integers (to be found experimentally or otherwise), which are probably some functions of n.&lt;/para&gt;For example, in cited by Lucas old known formula for&lt;para&gt;22/7 – Pi = Int(x^4*(1-x)^4*/(1+x^2),x = 0 .. 1)&lt;/para&gt; &lt;para&gt;generalization yields n=3, i=0, j=0, k=0, l=4, m=4&lt;/para&gt;In Lucas’s formula for&lt;para&gt;Pi – 333/106 = 1/530*Int(x^5*(1-x)^6*(197+462*x^2)/(1+x^2),x = 0 .. 1)&lt;/para&gt; &lt;para&gt;n=4, i=265, j=1, k=197, l=5, m=6&lt;/para&gt;In Lucas’s formula for&lt;para&gt;355/113 – Pi = 1/3164*Int(x^8*(1-x)^8*(25+816*x^2)/(1+x^2),x = 0 .. 1)&lt;/para&gt; &lt;para&gt;n=5, i=791, j=2, k=25, l=8, m=8&lt;/para&gt; &lt;para&gt;In Lucas’s formula for&lt;/para&gt; &lt;para&gt;Pi – 103993/33102 = 1/755216*Int(x^14*(1-x)^12*(124360-77159*x^2)/(1+x^2),x = 0 .. 1)&lt;/para&gt; &lt;para&gt;n=6, i= -47201, j=4, k=124360, l=14, m=12&lt;/para&gt; &lt;para&gt;In Lucas’s formula for&lt;/para&gt; &lt;para&gt;104348/33215 – Pi = 1/38544*Int(x^12*(1-x)^12*(1349-1060*x^2)/(1+x^2),x = 0 .. 1)&lt;/para&gt; &lt;para&gt;n=7, i= -2409, j=4, k=1349, l=12, m=12&lt;/para&gt; &lt;para&gt;&lt;/para&gt;.Thomas Baruchel has conducted extensive calculations and observed that in originally supplied five parameter notation (i,j,k,l,m)&lt;/para&gt; &lt;para&gt;j = m/2 – 2 and corespondingly m=2*(j+2)&lt;/para&gt; &lt;para&gt;This makes the original conjecture to depend on 4 parameters and to look like:&lt;/para&gt; &lt;para&gt;(-1)^n\cdot(\pi – \text{A002485}(n)/\text{A002486}(n))=(|i|\cdot2^j)^{-1} \int_0^1 \big(x^l(1-x)^{2(j+2)}(k+(i+k)x^2)\big)/(1+x^2)\; dx&lt;/para&gt; &lt;para&gt;and in unformatted form:&lt;/para&gt; &lt;para&gt;(-1)^n*(Pi−A002485(n)/A002486(n))=(abs(i)*2^j)^(-1)*Int((x^l*(1-x)^(2*(j+2))*(k+(i+k)*x^2))/(1+x^2),x=0…1)&lt;/para&gt; &lt;para&gt;Note that expanded expression, presented under the integral on the right hand side consists of 3 terms and looks like:&lt;/para&gt; &lt;para&gt;(k*(1-x)^(2*(j+2))*x^(l+2))/(x^2+1) + (k*(1-x)^(2*(j+2))*x^l)/(x^2+1) + (i*(1-x)^(2*(j+2))*x^(l+2))/(x^2+1)&lt;/para&gt; &lt;para&gt;Based on his calculations results, Thomas Baruchel also found that even with 4 parameters, this formula yields infinite number of solutions for each n.&lt;/para&gt; &lt;para&gt;And, of course, it would be nice to find some interesting rule on (l,m) in order to minimize (i,k) and, even better, it would be real cool to reduce the number of parameters just to one.&lt;/para&gt; &lt;para&gt;Thomas shared with me his calculations results and so now I have a lot of experimentally found five-tuples {n,i, j, k, l} – where n varies in the range from 2 to 26 and for each value of “n” Thomas supplied me with quite a few of valid combinations of i, j, k, l values … and based on this data, of course, it would be nice to find how (if at all) i, j, k, l are inter-related between each other and with “n” – but such inter-relation (if exists) is not obvious and difficult to derive just by observation … (though it is clearly seen that an absolute value of “i” is strongly increasing as “n” is growing from 2 to 26).<sec><heading></heading><para></para></sec>References:http://www.austms.org.au/Publ/Gazette/2005/Sep05/Lucas.pdfhttps://www.researchgate.net/publication/267998655_Integral_approximations_to_p_with_nonnegative_integrandshttp://web.maths.unsw.edu.au/~mikeh/webpapers/paper141.pdf