A197036 -id:A197036 - cp's OEIS frontend

A348607 Decimal expansion of BesselJ(1,2).

5, 7, 6, 7, 2, 4, 8, 0, 7, 7, 5, 6, 8, 7, 3, 3, 8, 7, 2, 0, 2, 4, 4, 8, 2, 4, 2, 2, 6, 9, 1, 3, 7, 0, 8, 6, 9, 2, 0, 3, 0, 2, 6, 8, 9, 7, 1, 9, 6, 7, 5, 4, 4, 0, 1, 2, 1, 1, 3, 9, 0, 2, 0, 7, 6, 4, 0, 8, 7, 1, 1, 6, 2, 8, 9, 6, 1, 2, 1, 8, 4, 9, 4, 8, 3, 9, 9

Offset: 0

Dumitru Damian, Oct 25 2021

			0.5767248077568733872...

Cf. A010790, A054687, A058798, A058797, A133920, A245308, A296168, A301484, A318224.

Bessel function values: A334380 (J(0,1)), A091681 (J(0,2)), A334383 (J(0,sqrt(2))), this sequence (J(1,2)), A197036 (I(0,1)), A070910 (I(0,2)), A334381 (I(0,sqrt(2))), A096789 (I(1,2)).

RealDigits[BesselJ[1, 2], 10, 100][[1]] (* Amiram Eldar, Oct 25 2021 *)

besselj(1, 2) \\ Michel Marcus, Oct 25 2021

Sage
```
bessel_J(1, 2).n(digits=100)
```

Equals Sum_{k>=0} (-1)^k/(k!*(k+1)!).

A334378 Decimal expansion of Sum_{k>=0} 1/((2*k+1)!)^2.

Original entry on oeis.org

1, 0, 2, 7, 8, 4, 7, 2, 6, 1, 5, 9, 7, 4, 1, 5, 7, 9, 9, 6, 9, 2, 6, 8, 8, 4, 9, 3, 0, 8, 0, 7, 9, 2, 3, 6, 3, 7, 3, 0, 3, 4, 3, 3, 1, 0, 2, 8, 3, 4, 2, 5, 7, 2, 5, 4, 7, 1, 2, 4, 5, 0, 2, 2, 8, 2, 6, 7, 2, 5, 6, 9, 2, 7, 3, 2, 3, 3, 2, 8, 1, 8, 8, 5, 7, 3, 5, 2, 7, 8, 8, 3, 5, 1, 5, 2, 8, 2, 6, 6, 4, 6, 7, 6, 7, 9, 2, 3, 7, 8

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/1!^2 + 1/3!^2 + 1/5!^2 + 1/7!^2 + ... = 1.027847261597415799692...
Continued fraction: 1 + 1/(36 - 36/(401 - 400/(1765 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (2*n*(2*n + 1))^2 for n >= 1. - _Peter Bala_, Feb 22 2024

Index entries for sequences related to Bessel functions or polynomials

Cf. A009445, A049469, A070910, A073742, A091681, A197036, A334379.

RealDigits[(BesselI[0, 2] - BesselJ[0, 2])/2, 10, 110] [[1]]

suminf(k=0, 1/((2*k+1)!)^2) \\ Michel Marcus, Apr 26 2020

(besseli(0,2) - besselj(0,2))/2 \\ Michel Marcus, Apr 26 2020

Equals (BesselI(0,2) - BesselJ(0,2))/2.

A367729 Decimal expansion of BesselI(0,2/sqrt(3)).

Original entry on oeis.org

1, 3, 6, 2, 1, 6, 1, 6, 3, 9, 6, 0, 9, 7, 8, 9, 9, 0, 4, 9, 4, 3, 1, 4, 3, 6, 2, 8, 4, 1, 4, 5, 5, 0, 0, 7, 3, 1, 8, 4, 0, 4, 3, 5, 4, 2, 9, 0, 0, 1, 3, 1, 5, 8, 7, 0, 7, 0, 2, 3, 6, 5, 7, 6, 4, 0, 0, 5, 5, 5, 6, 3, 6, 7, 8, 7, 8, 7, 8, 4, 6, 7, 1, 9, 1, 3, 0, 0, 9, 8, 1, 4, 9, 8, 3, 5, 2, 7

Offset: 1

Ilya Gutkovskiy, Nov 28 2023

			1.36216163960978990494314362841455...

Cf. A070910, A092041, A197036, A334381, A367730.

RealDigits[BesselI[0, 2/Sqrt[3]], 10, 98][[1]]

besseli(0,2/sqrt(3)) \\ Michel Marcus, Nov 29 2023

Equals Sum_{k>=0} 1 / (3^k * k!^2).

A212186 Decimal expansion of the integral over exp(x)/sqrt(1-x^2) dx between 0 and 1.

Original entry on oeis.org

3, 1, 0, 4, 3, 7, 9, 0, 1, 7, 8, 5, 5, 5, 5, 5, 0, 9, 8, 1, 8, 1, 7, 6, 9, 8, 6, 3, 1, 8, 7, 7, 9, 4, 7, 6, 7, 2, 2, 8, 9, 0, 9, 2, 0, 3, 3, 6, 1, 3, 6, 8, 3, 5, 0, 9, 7, 2, 4, 8, 8, 8, 2, 6, 1, 9, 6, 8, 1, 4, 0, 3, 2, 6, 9, 9, 3, 9, 9, 9, 5, 8, 0, 2, 7, 8, 4, 6, 5, 6, 6, 3, 6, 1, 4, 8, 3, 9, 7, 6, 5, 8, 2, 8, 1, 1, 9

Offset: 1

R. J. Mathar, Feb 13 2013

This appears as the first integral in an attempt to expand exp(x) in a Chebyshev series between 0 and 1. Other integrals of the higher order terms in that expansion are generally bootstrapped from the lower order terms.

If we substitute x=cos(y), this is the integral over exp(cos(y)) dy from y=0 to y=Pi/2, which matches (apart from the upper limit) eq. 3.915.4 of the Gradsteyn-Ryzhik tables. - R. J. Mathar, Feb 15 2013

			3.104379017855555098181769863187794767228...

A. Karageorghis, A note on the Chebyshev coefficients of the general order derivative of an infinitely differentiable function, J. Comp. Appl. Math 21 (1988) 129-132.

RealDigits[ Pi*(BesselI[0, 1] + StruveL[0, 1])/2, 10, 107] // First (* Jean-François Alcover, Feb 21 2013 *)
RealDigits[Integrate[Exp[x]/Sqrt[1-x^2],{x,0,1}],10,120][[1]] (* Harvey P. Dale, Jul 05 2025 *)

Equals Pi*(A197036+A197037)/2 .

A226975 Decimal expansion I_1(1), the modified Bessel function of the first kind.

Original entry on oeis.org

5, 6, 5, 1, 5, 9, 1, 0, 3, 9, 9, 2, 4, 8, 5, 0, 2, 7, 2, 0, 7, 6, 9, 6, 0, 2, 7, 6, 0, 9, 8, 6, 3, 3, 0, 7, 3, 2, 8, 8, 9, 9, 6, 2, 1, 6, 2, 1, 0, 9, 2, 0, 0, 9, 4, 8, 0, 2, 9, 4, 4, 8, 9, 4, 7, 9, 2, 5, 5, 6, 4, 0, 9, 6, 4, 3, 7, 1, 1, 3, 4, 0, 9, 2, 6, 6, 4, 9, 9, 7, 7, 6, 6, 8, 1, 4, 4, 1, 0, 0, 6, 4, 6, 7, 7, 8, 8, 6

Offset: 0

Horst-Holger Boltz, Jun 25 2013

This is also the derivative of the zeroth modified Bessel function at 1.

			0.56515910399248502720769602760986330732889962162109...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 51, page 504.

Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.

Cf. A002454, A197036.

Mathematica
```
RealDigits[BesselI[1, 1], 10, 110][[1]]
```

besseli(1,1) \\ Charles R Greathouse IV, Feb 19 2014

((1/2) * sum(1 / (4^x * factorial(x) * rising_factorial(2, x)), x, 0, oo)).n(360)
# Peter Luschny, Jan 29 2024

From Antonio Graciá Llorente, Jan 29 2024: (Start)
I_1(1) = (1/2) * Sum_{k>=0} (2*k)/(4^k*k!^2) = (1/2) * Sum_{k>=0} (2*k)/A002454(k).
Equals (1/2) * Sum_{k>=0} (4*k^2 + 4*k - 1) / (2*k)!!^2.
Equals exp(-1) * Sum_{k>=0} binomial(2*k,k+1)/(2^k*k!).
Equals (-e) * Sum_{k>=0} (-1/2)^k * binomial(2*k,k+1)/k!
Equals (1/Pi)*Integral_{t=0..Pi} exp(cos(t))*cos(t) dt. (End)

cp's OEIS Frontend

A348607 Decimal expansion of BesselJ(1,2).

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A334378 Decimal expansion of Sum_{k>=0} 1/((2*k+1)!)^2.

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A367729 Decimal expansion of BesselI(0,2/sqrt(3)).

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A212186 Decimal expansion of the integral over exp(x)/sqrt(1-x^2) dx between 0 and 1.

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A226975 Decimal expansion I_1(1), the modified Bessel function of the first kind.

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