What does this lambda–sigma notation mean?

I am currently reading the CRC Handbook of Phosphorus-31 Nuclear Magnetic Resonance Data by John C. Tebby (CRC Press, 1991) and on several figures (ex. pages 9 to 14) there is a caption that reads along these lines

$\ce{^{31}P}$ NMR chemical shifts of three coordinate (λ3 σ3) phosphorus compounds. (Tables B to E.)

What does the (λ3 σ3) signify? I've not seen that notation before, and it is on the captions of some of the charts I need.

Answer

The notation refers to the valency (number of valence electrons involved in bonds, i.e., either 3 or 5) and coordination number (number of substituents attached) in organophosphorus compounds. For example, it could refer to the tautomeric forms of phosphonate esters:¹

So the λ³σ³ refers to a P with 3 bonding valence electrons (the lone pair doesn't count) and 3 substituents attached while λ⁵σ⁴ refers to a P with all five of its valence electrons involved in bonds and 4 substituents attached.

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Reference

1. Kraszewski, A.; Stawinski, J. H-Phosphonates: Versatile synthetic precursors to biologically active phosphorus compounds. Pure Appl. Chem. 2009, 79 (12), 2217–2227. DOI: 10.1351/pac200779122217.