- #1

uart

Science Advisor

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I was looking at finding a series solution to a 2nd order DE the other day and came up with the following (for one of the solutions, and there was a somewhat similar series for the other solution).

[tex]\sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!} \prod_{m=1}^{k-1} (3m+1)[/tex]

Wolfram said the solutions were the Airy functions [itex]Ai[/itex] and [itex]Bi[/itex], and since these can be defined in terms of the Hypergeometric function, [itex]~_0F_1[/itex], I suspected that the series I had might be equivalent to a hypergeometric. http://mathworld.wolfram.com/AiryFunctions.html

Wolfram lists one of the hypergeometrics involved in [itex]Ai[/itex] and [itex]Bi[/itex] as, [itex]~_0F_1(\frac{2}{3}, \frac{x^3}{9})[/itex]. And when I expand this as per the definition here http://mathworld.wolfram.com/ConfluentHypergeometricLimitFunction.html I get the following.

[tex]\sum_{k=0}^{\infty} \frac{x^{3k}}{9^k \, k!} \left[ \prod_{m=0}^{k-1} (\frac{2}{3}+m) \right]^{-1}[/tex]

When I sum these numerically in matlab I seem to get the same answer for both. I can see a lot of similarity but can't quite make out the equality of the two. Can anyone see any easy way to show they're equal?

[tex]\sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!} \prod_{m=1}^{k-1} (3m+1)[/tex]

Wolfram said the solutions were the Airy functions [itex]Ai[/itex] and [itex]Bi[/itex], and since these can be defined in terms of the Hypergeometric function, [itex]~_0F_1[/itex], I suspected that the series I had might be equivalent to a hypergeometric. http://mathworld.wolfram.com/AiryFunctions.html

Wolfram lists one of the hypergeometrics involved in [itex]Ai[/itex] and [itex]Bi[/itex] as, [itex]~_0F_1(\frac{2}{3}, \frac{x^3}{9})[/itex]. And when I expand this as per the definition here http://mathworld.wolfram.com/ConfluentHypergeometricLimitFunction.html I get the following.

[tex]\sum_{k=0}^{\infty} \frac{x^{3k}}{9^k \, k!} \left[ \prod_{m=0}^{k-1} (\frac{2}{3}+m) \right]^{-1}[/tex]

When I sum these numerically in matlab I seem to get the same answer for both. I can see a lot of similarity but can't quite make out the equality of the two. Can anyone see any easy way to show they're equal?

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