## Wednesday, March 10, 2021

### [uekvaxju] A052119 constant

the number whose continued fraction expansion is [0,1,2,3, ...] is an elegant constant with no official name.  it is the ratio of incomplete Bessel functions:

? x=besseli(1,2)
1.5906368546373290633822544249996662479544781594955366471322879846085450375353611851161221475942289252377564135042805648221333978893456722913616450824073717776541321837633334921725227835116047303290497

? y=besseli(0,2)
2.2795853023360672674372044408115333532858411027854590540708397516643053432326763427295170885564858989845955206368381095580572209149909049855156258765330473402308430842678623161078222092234217628420350

? x/y
0.69777465796400798200679059255175259948665826299802123236863008281653085276464111299696565418267656872398282187739641339311319229611953258394826715402336857207708468793165325967680260969934477352791348

? contfrac(x/y)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 72]

the numerator and denominator do not have interesting simple continued fraction expansions.

? contfrac(x)
[1, 1, 1, 2, 3, 1, 6, 1, 4, 2, 5, 10, 21, 21, 1, 12, 1, 2, 1, 1, 1, 1, 1, 2, 3, 2, 3, 5, 5, 1, 2, 5, 1, 1, 6, 24, 2, 1, 15, 1, 3, 1, 3, 1, 3, 18, 15, 3, 1, 1, 3, 2, 1, 2, 3, 1, 3, 1, 2, 21, 1, 1, 4, 3, 7, 7, 2, 1, 5, 1, 2, 6, 4, 1, 4, 4, 2855, 1, 10, 1, 8, 1, 1, 1, 165, 22, 8, 5, 7, 2, 7, 1, 2, 1, 1, 1, 1, 1, 12, 4, 1, 1, 2, 1, 15, 1, 3, 5, 1, 1, 6, 8, 6, 1, 3, 1, 9, 4, 2, 27, 1, 1, 7, 1, 11, 2, 1, 2, 1, 1, 6, 23, 6, 1, 8, 4, 1, 1, 4, 2, 3, 1, 1, 68, 1, 1, 2, 3, 5, 25, 1, 6, 1, 3, 1, 1, 1, 2, 9, 1, 9, 4, 3, 3, 10, 1, 3, 3, 1, 1, 1, 2, 1, 1, 4, 4, 2, 1, 4, 1, 2, 1, 8, 2, 2, 16, 1, 1, 3, 10, 2, 1, 1, 4, 1, 1, 2, 439, 1, 3, 1, 2, 21, 5, 11]

? contfrac(y)
[2, 3, 1, 1, 2, 1, 3, 7, 4, 3, 1, 2, 2, 1, 2, 1, 1, 2, 7, 8, 1, 1, 21, 1, 16, 2, 1, 8, 1, 1, 8, 1, 35, 1, 2, 1, 1, 4, 1, 1, 1, 3, 132, 3, 1, 10, 2, 1, 1, 1, 1, 2, 2, 6, 100, 1, 1, 26, 1, 66, 1, 2, 16, 1, 4, 52, 2, 1, 1, 1, 16, 8, 1, 3, 172, 1, 3, 1, 3, 3, 1, 13, 1, 5, 2, 1, 4, 3, 1, 3, 4, 7, 1, 1, 23, 1, 1, 2, 5, 8, 2, 1, 3, 3, 1, 2, 10, 2, 1, 1, 1, 1, 2, 1, 1, 1, 25, 1, 1, 1, 39, 15, 2, 9, 1, 5, 1, 1, 1, 2, 31, 3, 12, 2, 1, 6, 1, 2, 1, 1, 1, 3, 1, 12, 6, 2, 1, 1, 1, 2, 2, 2, 1, 10, 2, 3, 2, 1, 24, 1, 2, 1, 10, 1, 1, 10, 1, 1, 2, 2, 4, 5, 1, 3, 5, 8, 1, 2, 1, 440, 4, 2, 1, 5, 2, 1, 11, 2, 3, 1, 15, 1, 25, 1, 4, 1, 8, 2, 8, 2, 159, 8, 1, 4, 7, 2]

related sequences linked from https://oeis.org/A052119 note that there are generalized continued fraction expansions of the numerator and denominator which do have structure.

above computations are with Pari/GP.  because of round off error, some number of final terms are incorrect.  i think Mathematica does a better job at making sure that all output continued fraction terms of an input symbolic expression are correct.