Virahanka

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Virahanka (विरहाङ्क) was an Indian prosodicist who is also known for his work on mathematics. He possibly lived in the 6th century AD, but it is also possible that this date may be as late as 8th century. His work on prosody builds on the Chhanda-sutras of Pingala, and was the basis for a 12th c. commentary by Gopala.

[edit] Contribution to Fibonacci Series

In an analysis of mAtrA-vrittas - combinations that obtain certain meters or mAtrAs, Virhanka analyzes a mixture of short (laghu) and long (guru) syllables that appear in a meter.

For example, the gAyAtri mantra contains \the string sa-vi-tur-va-reN-yam, in which tur and reN are long or guru, the others being short or laghu. Using G and L for guru and laghu, we can write this meter as LLGLGL, which is the kernel of the famous gAyAtri chhanda.

In the mAtrA-vritta analysis, G is taken to be twice as long as an L syllable, and Virahanka considers the question: in how many ways can we form meters of length n?

The answer emerges as follows:

Length 1: L (1)
Length 2: G, LL (2)
Length 3: LG, GL, LLL (3)
Length 4: GG, LLG, LGL, GLL, LLLL (5)
Length 5: LGG, GLG, LLLG, GGL, LLGL, LGLL, GLLL, LLLLL (8)

If we denote S(n) as the number of ways in which a meter can be of length n, then we see that S(n+1) is obtained by adding a G at the end for all the S(n-1) strings and an L at the end for all the S(n) strings. Therefore S(n+1) = S(n)+S(n-1).

This is the well known Fibonacci sequence. Thus, Virahanka has at least four centuries of primacy over Fibonacci.

Subsequently commentaries by Gopala elaborated further on the sequence that now bears the name of Fibonacci. A 14th century text, Prakrit-Paingala, also gives a formula for computing fibonacci numbers in terms of binomial coefficents.