Re: [math-fun] Atrocious function plots
Fred> A short while back, I described a purported plot of the classical Beta function at http://mathworld.wolfram.com/BetaFunction.html --- or maybe it was the the very similar offering at https://en.wikipedia.org/wiki/Beta_function --- as "passable identikit". Given that identikit reconstructions are notorious for bearing at best only a remote resemblance to their subjects, the verdict may well have been more appropriate than I could known at the time. The sole noteworthy feature of these hilarious travesties proves to be merely a lattice of vertical planar artifacts, resulting from a misguided attempt to plot across lines of discontinuity, and completely obliterating any structural detail of the actual surface. In the course of drafting an account of the recent discussion of binomial coefficients in this list, I took up Gosper's challenge to produce an accurate plot of the associated function of two real variables; and while I was at it knocked out a plot of the (closely related) Beta function, which I have now posted at https://www.dropbox.com/s/rzcojf8wibd0mpw/beta5.png Compare the competing versions; and weep. Fred Lunnon -------------- I hope Fred will soon announce his lovely new pdf that combines enhancements of these plots with compelling mathematics supporting Binomial[-n,-k]:=0 . Here's one I just thought of. For nonnegative integer n In[12]:= Sum[Binomial[n - k, k], {k, 0, u}] Out[12]= Fibonacci[1 + n] for Floor[n/2] ≤ u ≤ n . For u>n, we crash into the 2 o'clock wedge and tack on (-1)^n times an interesting integer depending on how far we transgress. For u=∞, the sum diverges. But we can dispel all this ugliness by merely left-right flipping the sum trajectory: In[12]:= Sum[Binomial[n - k, n-2*k], {k, 0, u}] Out[12]= Fibonacci[1 + n] for all u ≥ Floor[n/2], including ∞, because the 10 o'clock wedge (which Fred calls the Bermuda Triangle) isn't there. Playing with this in Mma, I encountered such a swarm of bugs that I don't know where to begin reporting them. E.g., I could get √5 as a finite sum of integers. The situation is nearly irreparable due Mma's inconsistent and incomplete Sum semantics, and to the depressing fact that pFq notation itself is broken. E.g., Hypergeometric1F1[1-n,-n,n]==Sum[n^(i-1)*(n-i)/i!,{i,0,∞}]==0 unless n is a nonnegative integer, when Hypergeometric1F1[1-n,-n,n]==Sum[n^(i-1)*(n-i)/i!,{i,0,n}]==n^n/n! In other words, there is the insane convention that the hypergeometric series terminates if a random term happens to vanish, regardless of ensuing terms. Thus In[25]:= FunctionExpand[Hypergeometric1F1[1-n,-n,n]] Out[25]= 0 is improvident, because In[28]:= Hypergeometric1F1[1-n,-n,n]/.n->2 Out[28]= 2 Oh well, another reason to switch to matrix products. --rwg
Replacing the Binomials with QBinomials in the Fibonacci identities below, and throwing in a factor of q^k^2, we get the "q-Fibonacci numbers": In[47]:= Expand[FunctionExpand[Table[Sum[q^k^2*QBinomial[n-k,k,q],{k,0,n}],{n,{0,1,2,3,4,5,6,19}}]]] Out[47]= {1,1,1+q,1+q+q^2,1+q+q^2+q^3+q^4,1+q+q^2+q^3+2 q^4+q^5+q^6,1+q+q^2+q^3+2 q^4+2 q^5+2 q^6+q^7+q^8+q^9,1+q+q^2+q^3+2 q^4+2 q^5+3 q^6+3 q^7+4 q^8+5 q^9+6 q^10+7 q^11+9 q^12+10 q^13+12 q^14+14 q^15+17 q^16+19 q^17+23 q^18+25 q^19+29 q^20+32 q^21+37 q^22+40 q^23+46 q^24+50 q^25+56 q^26+61 q^27+68 q^28+73 q^29+81 q^30+86 q^31+94 q^32+100 q^33+108 q^34+113 q^35+122 q^36+127 q^37+135 q^38+140 q^39+148 q^40+152 q^41+160 q^42+163 q^43+169 q^44+172 q^45+177 q^46+177 q^47+182 q^48+181 q^49+183 q^50+181 q^51+182 q^52+178 q^53+178 q^54+172 q^55+170 q^56+164 q^57+160 q^58+152 q^59+148 q^60+139 q^61+133 q^62+124 q^63+118 q^64+108 q^65+102 q^66+92 q^67+85 q^68+76 q^69+70 q^70+61 q^71+56 q^72+48 q^73+43 q^74+36 q^75+32 q^76+26 q^77+23 q^78+18 q^79+15 q^80+12 q^81+10 q^82+7 q^83+6 q^84+4 q^85+3 q^86+2 q^87+2 q^88+q^89+q^90} In[48]:= %/.q->1 Out[48]= {1,1,2,3,5,8,13,6765} (6765 was the console phone # at MIT AI.) These polynomials approach a limit, q-Fibonacci(∞) which is the Rogers-Ramanujan function: In[55]:= Series[1/Product[(1-q^(5*n-4))*(1-q^(5*n-1)),{n,∞}],{q,0,36}] Out[55]= 1+q+q^2+q^3+2 q^4+2 q^5+3 q^6+3 q^7+4 q^8+5 q^9+6 q^10+7 q^11+9 q^12+10 q^13+12 q^14+14 q^15+17 q^16+19 q^17+23 q^18+26 q^19+31 q^20+35 q^21+41 q^22+46 q^23+54 q^24+61 q^25+70 q^26+79 q^27+91 q^28+102 q^29+117 q^30+131 q^31+149 q^32+167 q^33+189 q^34+211 q^35+239 q^36+O[q]^37 (The sum (with its upper limit infinite, with or without reflecting the q-binomial) is not obviously the usual one in the Rogers-Ramanujan identity.) The coefficient sequence A003114 <https://oeis.org/A003114>has a bunch of interesting combinatorial interpretations, enriching the property list of 239 (if you accept its index 36 as somehow fundamental). --rwg On Sun, Aug 4, 2013 at 12:56 AM, Bill Gosper <billgosper@gmail.com> wrote:
Fred> A short while back, I described a purported plot of the classical Beta function at http://mathworld.wolfram.com/BetaFunction.html --- or maybe it was the the very similar offering at https://en.wikipedia.org/wiki/Beta_function --- as "passable identikit".
Given that identikit reconstructions are notorious for bearing at best only a remote resemblance to their subjects, the verdict may well have been more appropriate than I could known at the time. The sole noteworthy feature of these hilarious travesties proves to be merely a lattice of vertical planar artifacts, resulting from a misguided attempt to plot across lines of discontinuity, and completely obliterating any structural detail of the actual surface.
In the course of drafting an account of the recent discussion of binomial coefficients in this list, I took up Gosper's challenge to produce an accurate plot of the associated function of two real variables; and while I was at it knocked out a plot of the (closely related) Beta function, which I have now posted at https://www.dropbox.com/s/rzcojf8wibd0mpw/beta5.png
Compare the competing versions; and weep.
Fred Lunnon -------------- I hope Fred will soon announce his lovely new pdf that combines enhancements of these plots with compelling mathematics supporting Binomial[-n,-k]:=0 .
Here's one I just thought of. For nonnegative integer n
In[12]:= Sum[Binomial[n - k, k], {k, 0, u}]
Out[12]= Fibonacci[1 + n]
for Floor[n/2] ≤ u ≤ n . For u>n, we crash into the 2 o'clock wedge and tack on (-1)^n times an interesting integer depending on how far we transgress.
For u=∞, the sum diverges.
But we can dispel all this ugliness by merely left-right flipping the sum trajectory: In[12]:= Sum[Binomial[n - k, n-2*k], {k, 0, u}]
Out[12]= Fibonacci[1 + n]
for all u ≥ Floor[n/2], including ∞, because the 10 o'clock wedge (which Fred
calls the Bermuda Triangle) isn't there.
Playing with this in Mma, I encountered such a swarm of bugs that I don't know where to begin reporting them. E.g., I could get √5 as a finite sum of integers.
The situation is nearly irreparable due Mma's inconsistent and incomplete Sum semantics, and to the depressing fact that pFq notation itself is broken. E.g., Hypergeometric1F1[1-n,-n,n]==Sum[n^(i-1)*(n-i)/i!,{i,0,∞}]==0
unless n is a nonnegative integer, when Hypergeometric1F1[1-n,-n,n]==Sum[n^(i-1)*(n-i)/i!,{i,0,n}]==n^n/n! In other words, there is the insane convention that the hypergeometric series terminates if a random term happens to vanish, regardless of ensuing terms.
Thus In[25]:= FunctionExpand[Hypergeometric1F1[1-n,-n,n]] Out[25]= 0 is improvident, because In[28]:= Hypergeometric1F1[1-n,-n,n]/.n->2 Out[28]= 2 Oh well, another reason to switch to matrix products.
--rwg
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Bill Gosper